There's a great book called The Rise of Modern Japan by Peter Duus, and though it deals with the modernization and industrialization of Japan, it does touch on the tenuous relationship between emperor and shogun. He writes "there was no culturewide political authority able to guarantee the public order, nor any political center to which all Japanese gave their undivided political loyalty... ....at the very highest level, political leadership was divided between two authorities bound together in a symbiotic relationship." Duus describes the Emperor as a distant and mythic figure to the common people; a "semidivine civil monarch." The shogun was the "feudal suzeraine" of the hundreds of daimyo and served as a "military monarch." He describes their relationship as follows: "In theory, the shogun was the deputy of the emperor's authority, exercising the sovereign powers that the emperor had long since lost. In fact, the shogun was the patron of the emperor, for the Tokugawa family had in the early seventeenth century granted the imperial family landed income to rescue it from poverty and enable it to maintain a court and aristocratic retinue." I hope this helps give you some perspective in answering your question!

TL;DR, it depends on whether the request was political or not. If it was not political, the Emperor's request would likely be fulfilled, at least as much as was possible. If it was, then it depends on the period.

Part 1: Kamakura to before the Sengoku

The power of the emperor vis-a-vis the Shōgun waxed and waned throughout the ages. So let’s give it a run down of the political history.

The first generalization that we need to correct is that the emperor did not become figureheads, so to speak, when the Bakufu was formed in 1185. It held considerable political and administrative power during the Kamakura. In fact, one could easily argue that the court had more power, and the Bakufu was the junior party of the arrangement. The land-owning rights and administrative head of the provinces were all still belonged to the officials of the court. The Bakufu only had power to appoint the jitō, who collected taxes and administered the land, and shugo, who was in charge of military and policing, in Eastern Japan only. All of western Japan was in effect still under control of the Kyōto court. It was not until retired-emperor Go-Toba’s disastrous attempt at destroying the Kamakura Bakufu in 1221 (the Jōkyū War), that Kamakura Bakufu was able to appoint jitō and shugo to the lands of those who had supported Go-Toba and were punished. Imperial succession also had to go through the Bakufu. However, considering the emperor still held sway over some parts of Japan, and the land rights and administrative head were courtiers, the emperor still was not a figurehead, just the (significantly) weaker half of a dyarchy. In the aftermath of the Jōkyū War, the Bakufu gained the power to have all complaints about jitō and shugo (i.e. when they were overstepping their powers) go through Kamakura, which allowed them to gradually strip rights from courtiers and crown in the provinces, but it wasn’t until the needs of the Mongol invasion that the court gave up what’s left of it’s provincial military mobilization powers to Kamakura.

Ironically, while the emperor slowly became a figurehead after the Jōkyū War, the Shōgun was already not only a figurehead, but also a hostage. The death Kamakura Bakufu’s founder Minamoto Yoritomo in 1199 began a tragic power struggle within the Bakufu that lead to the deposing and assassination of the second Shōgun Yoriie in 1203 and 1204 respectively, and the assassination of the third Shōgun Sanetomo in 1219. With that Yoritomo’s line ended. The Hōjō (Yoritomo’s in-laws), who had became strongest power in Bakufu asked Go-Toba for an imperial prince to be Shōgun. Go-Toba refused, instead appointing a child from the regent family Fujiwara who had Minamoto blood, to be the Hōjō’s puppet, so you can see the throne still had power to decide who was Shōgun at this point. Also due to this string of disturbance, Go-Toba mistakenly thought that the Kamakura Bakufu could be easily crushed, leading to the Jōkyū War, after which the power structure flipped. But in any case, the fourth and fifth Kamakura Shōguns were from the Fujiwara, and the sixth to ninth Shōguns were imperial princes. This both demonstrate the weakening of the power of the crown following the Jōkyū War, while also showing that, in effect the Shōguns themselves were far worse off, being puppets and hostages from the court to the Hōjō.

In 1333, (then ex-) Emperor Go-Daigo rose up against the Kamakura Bakufu. The Bakufu, weakened from the strain of the defense against the Mongols and internal in-fighting, was destroyed as many powerful vassals switched sides to the Emperor. For the next few years, Go-Daigo restored direct rule to the country, before Ashikaga Takauji, who had helped Go-Daigo defeat the Kamakura, turned on Go-Daigo in 1335 and made himself Shōgun and enthroned a puppet emperor in 1336. Go-Daigo himself, his descendants and those loyal to them continued to resist until 1392. This is known as the period of the Northern and Southern Courts (Nanbokuchō)

Here we can see that the Emperor’s position as head of the Japanese was so solid that Ashikaga Takauji did not dare naming himself emperor outright, even when he ran the reigning emperor out of Kyōto. Even Takauji could only name another prince as emperor. On the other hand, the loyalty of most warrior families was fickle. The emperor’s words had authority and legitimacy, and if his interests align with those of the warrior families, they would follow him. But if those interests don’t align, they wouldn’t. In this time, the emperor of the Northern Court, under the Ashikaga, were without-a-doubt puppets. Those of the Southern headed the resistance, and while they depended on important warrior families like the Kitabatake to carry out their wishes, they weren’t figureheads.

The emperor was probably in its weakest in all of the Bakufu era in the years immediately following 1392. It has no military. All external political functions are under the Muromachi Bakufu. While actual crown lands are sort of secure, the manors of courtiers in the provinces have been taken under wing by the shugo-daimyō, who, asides from military and policing, has taken over administration, tax-collection, and was entitled to half of the tax revenue from manors. On top of that, not only did the third Muromachi Shōgun Yoshimitsu take over religious appointments, court appointments, and the power to give out official ranks and titles, but Yoshimitsu also acted as a retired emperor would at court. He (then retired-Shōgun, but holding on to power) got himself named Chancellor of the Realm (dajō-daijin), got the young Emperor Komatsu to marry his wife’s adopted sister, and got his wife named Komatsu’s godmother. When he died in 1408, the court actually bestowed on him the title of Retired Emperor.

After Yoshimitsu’s death, the Bakufu took a step back interfering with the emperor and the court, and let the court resume its non-political duties prior to Yoshimitsu’s takeover (the Bakufu kept the right to listen to petitions for ranks and titles, acting as the middle-man) Still, it had no function outside the Kyōto court, and was more of a cultural/religious/spiritual head than anything political. This began to change as the Ashikaga Bakufu encountered increasingly frequent local warfare, uprisings, and rebellions. In 1438, the Ashikaga Yoshinori came to the Emperor Go-Hanazono and asked him to issue the official edict to name the Eikyō rebellion enemies of the throne to be eliminated. Called the chibatsu rinji, these edicts haven’t been issued since the time of Yoshimitsu’s heyday, and shows if not a weakening of the Bakufu, at least an increasingly desperate, such that the Bakufu finally had to, once again, ask the emperor to lend his legitimacy to the Bakufu’s cause. If the Bakufu wasn’t weakening under Yoshinori, it certainly greatly weakened after he was assassinated in 1441, after which the chibatsu rinji became quite regular. This points out an important thing: while the emperor and court by now had no military or administrative powers, even the Bakufu recognize that the emperor’s orders offered strong legitimacy. When the Bakufu was strong, it preferred not to rely on the emperor, but as it weakened, it had no choice but to do so. The weaker the Bakufu, the more important imperial support became, and this trend continued into the Sengoku.

Before going into the Sengoku, I’d like to just quickly mention the so called “Later-Southern Court”. Yoshimitsu originally brought the Nanbokuchō to an end by promising alternating imperial succession, but Emperor Komatsu, with the Bakufu’s agreement, broke the promise and passed the throne to his son instead of the prince from the Southern Court. This caused periodic rebellion. The biggest of these occurred in 1443, when men loyal to the Southern Court attacked the palace, tried to assassinate the Emperor Go-Hanazono, failed but got away with the Three Sacred Treasures. While these rebellions came to nothing, and the Treasures were eventually restored, it demonstrated how powerful the imperial name could be.

I think one good incident that sort of answers all of your questions was the "Purple Robe Incident" of of 1627. In 1615, the Tokugawa bakufu enforced an edict that relegated the position of the Emperor to that of a ceremonial figurehead. Many of the proclamations made regarding his behavior was that he must at all times get Shogunal authorization for certain actions that were previously entirely under the Emperor's purview.

One such action was that of bestowing purple robes onto Buddhist monks--an honorific act that gave the monks great prestige. The bakufu of course would want to break such ties and have complete control over this business, and so declared that the Emperor must get approval prior to bestowing the robes. However, Emperor Go-Mizunoo ignored this, and gave a number of priests purple robes in 1627. When the bakufu caught wind of this, they invalidated the robes and sent at least six of those priests into exile, severely embarrassing and undermining the Emperor's authority. Two years later in 1629, Mizunoo would abdicate and Tokugawa Ieyasu's great granddaughter would come to the throne.

I believe that this incident shows that the Emperor's authority was severely limited and the Imperial house was subordinate to the bakufu. Though the shogunate certainly used the Imperial house to legitimize themselves in the early days and, as stonedbearamerica stated, were patrons that kept the house afloat, they were by no means underneath the Emperor and instead used these to make the Emperor dependent on the shogunate. I would not say that this was new either, as medieval daimyo used the same strategies to position themselves as potential hegemons.

For instance, some of Princeton Professor Thomas Conlan's newer research shows that the Ouchi clan in Medieval times were planning on transferring the court from Kyoto to Hiroshima and were able to do so because of the severe state of poverty the court was in. By filling the coffers of the court and paying for the ceremonial events that gave the court its prestige, the court became indebted to the clan. After the fall of the Ouchi, other clans would continue this strategy, though not to the same degree until the rise of the Tokugawa bakufu.

If you would like to read more about the purple robe incident, Duncan Williams has an article entitled "The Purple Robe Incident and the Formation of the Early Modern Sōtō Zen Institution"

Why military needed to keep this strange two power centered arrangements for centuries. Here is China with dynasties falling and being established again. Tokugava could simply dipose emperor and become emperors themselves.