TL;DR There's no easy way to find out this information for JR East shinkansen lines. Information for the Kyushu/Sanyo/Tokaido shinkansen lines is easier to find though.
I remember seeing trainset types listed in printed timetables for sale in bookstores (and sometimes in the station) in Japan. Here's an example schedule, JTB時刻表, on Amazon Japan. You can see the row in the image below that has 500, 700, N700, etc. showing the trainset type.
However, upon further research, it looks like the train type is primarily only shown for the Tokaido and Sanyo shinkansen lines. You can find some schedules with trainset types online on JR Central's website.
For the shinkansen lines operated by JR East, it looks like things aren't so straight-forward. You can kind of tell by train type what it is, but there's unfortunately still some ambiguity. It sounds like with the schedule books you might be able to look towards the back where they have drawings of the trainsets to figure out what service uses what kind train, depending on the number of cars, etc. For example, if you know it's a Yamabiko + Tsubasa that's a total of 17 cars, it must be E2 + E3, if it's 16 cars, it must E5 + E3.
Definitely not easy though.
JR East's online timetable from Omiya towards Tokyo (in Japanese) shows which trains are combined and which trains they consist of (e.g. Yamabiko + Tsubasa).
I did, however, found this website (in Japanese) that lists some trainset types, but it does it line-by-line, so from Tokyo to Omiya you end up having to check multiple shinkansen line schedules. It's also hard to tell which ones are couples together (since they're usually on different lines).