Before the rise of the samurai, the Fujiwara clan held the position of empress, and from there was about to obtain significant power. For a significant period in the Heian period, they were the de facto ruler of Japan, controlling the regents position and appointing family members to important positions.

In the later Heian, though the position of regent remained with the Fujiwara, some emperors (especially those without Fujiwara mothers) began to assert their own authority. While the Fujiwara influence decreased at court, branches of the family became samurai. (EDIT: while no longer de facto ruler, the Fujiwara remained important and still produced a lot of imperial consorts)

With the rise of the samurai, the shogun became the one in charge of countrywide administration. The Kamakura though in name in the hands of the Minamoto, was actually in the hands of the Hojo. The Hojo's control started with Masako's marriage to Minamoto no Yoritomo, the first Kamakura shogun. From there they were able to secure the position of regent such that even when the main Minamoto line ended (less than 30 years after Yoritomo's appointment to shogun) the Hojo were the de facto ruler of the Kamakura shogunate, retaining their regency even when imperial princes were appointed shogun.

I believe from this point onwards the shogun (or in the case of the Kamakura, the Hojo regents) mostly married their daughters to branch family members or powerful samurai clans. As the emperor had long lost any actual power over the country, there isn't any reason but for prestige to marry your daughter to the emperor. Alliances with other powerful samurais were more important.

Source: William Faris. Japan to 1600. 2009

EDIT: Unsourced personal conjecture:

It would appear during the shogunate periods the vast majority of head imperial consorts were from the Fujiwara or one of the cadet branches. So the de jure status divide between the kuge (aristocrats) and buke (samurai), as well as tradition, might have made the position hard to penetrate. But then again, the shogun (or the shogun regent for the Hojo) was the de facto ruler anyway.

Going through the list of Japanese Emperors on Japanese wiki as reference, the times when the head consort was from a buke were:
a) twice during the Nanbokucho (Northern and Southern Dynasties) when Emperors of the southern court married women from the Kitabatake family. The southern court were emperors and their supporters fighting against the Ashikaga shoguns and the puppet emperors of the Northern court, and the Kitabatake were important supporters of the southern court.
b) once in the Edo bakufu when shogun Tokugawa Hidetada, the second Tokugawa shogun, married her daughter to the emperor. It probably gave a lot of prestige and legitimacy for the young shogunate.