The shogunate is an interesting topic, one I hope I can do justice to. The main shogun 将軍 is ultimately a derivative of the seii taishogun 征夷大将軍 you mentioned. Chinjufu shogun 鎮守府将軍 (aka chinju shogun) can also found as a separate title referring to the "Military General" under the seii-taishogun. To answer your question in short, yes there were campaigns against the emishi (毛人 or 蝦夷) to bring them into subjugation, which is ultimately where the seii-taishogun title comes from. But let's go back a bit further...

To understand the context, we must first know a bit about the formation of kingship and the Japanese State. I wrote a little about that in this response to another question, but critical to take away is the expansion of the Yamato people across Japan and the court's desire to bring unification amongst the various paramounts in the Yayoi, Kofun, Yamato/Asuka, and Nara eras. With more integration comes more codification, including the kani junikai 冠位十二階 (12 cap/rank system of Shotoku Taishi), the Taika Reforms (大化の改新 Taika no kaishin) under Emperor Kotoku in 645AD, the Omi-ryo 近江令 of 669AD, the Taiho Ritusryo 大宝律令 of 703AD, and Yoro Ritsuryo 養老律令 of 757AD. All were ritsuryo or codified laws modeled after Chinese laws.

Legal codification and "assimilation" of clans and paramounts could only go so far. The northeastern third of Honshu was still beyond the court's control in the 8th century. This area, sometimes referred to as michi no oku (道の奥, literally "deep in the road" but more like "the deep interior of the land"), was inhabited by what the Yamato called the emishi. Ainu may also be found in the area and people often interchange emishi and ainu, but for argument sake we'll use emishi for this tale. What would become known as the provinces of Mutsu and Dewa were beyond the court's control, were codified in the ritsuryo as being "outside the land that has been transformed" (kegai 化外), meaning outside the area "tranformed by" the court's control. The Nihon Shoki has tales of emish as "eastern savages" of "violent disposition" though modern scholarship knows the tales are different than reality. Still, that was the perception of the court, driven by cultural and geographical differences.

In the mid-7th century, the court started to penetrate the northeast by establishing outposts: first in Echigo, then in the following decades, sporadically into Dewa and Mutsu. These outposts were designed to create a Yamato presence on the land, not simply administer it like in the areas further south. Taga in Mutsu served as the provincial government HQ and the site of the Pacification HQ (Chinjufu 鎮守府). This is where the chinjufu shogun commanded with a staff of subordinates as defined in the ritsuryo.

The court tried dipolomacy. They sought to coopt emishi chieftons with invites to court to offer regional items as "tribute" in exchange for presents and/or court rank in return. Some were awarded kabane 姓 to help boost credence. Ritsuryo codes for governors in Mutsu and Dewa included specific directives to entertain emishi at banquets and present them with gifts- while spying on them the entire time. Emishi that came to terms with the state were sometimes subject to relocation to provinces further south, a strategy intended to give the court more control and to keep them from other emishi.

But the court kept pushing into emishi territory, becoming more intrusive and more destructive to their traditional lifestyles. Think of it as the European/American expansion vs the native Americans. Tensions rose, hostility occurred, and in 709 we see records of small-scale, conservative military action being taken, followed by a request for supplies "for the purpose of subjugating the barbarians" four months later. A few years later, murders of government officials in Mutsu prompted additional campaigns in 720 and 724. In 737, one of the better documented campaigns, Ono Azumabito, the chinju shogun and Mutsu governor created a task for road construction between Taga in Mutsu and the Dewa provincial HQ in Akita. A major incident with three emishi chieftons was avoided thanks to heavy snow and cool heads (pun not intended).

Little is known of activity after that until July 774 when an imperial decree announced the immediate dispatch of an army to "strike down and destory emishi resistance in a timely fashion." It was a significant policy shift done with the excuse that the "barbarians have no amended their wild hearts." The real reason(s) for this shift is not entirely known, but in 770 it's said an emishi chiefton Ukanume Ukutsunamio declared his intention to form a coalition of tribes and begin attacks on Yamato outposts, vanishing into the wilderness with his followers supposedly in preparation. What happened between then and July 774 we don't know other than in January 774 emishi and emishi who allied themselves with the Yamato (see above) were ordered to cease appearing at court.