Upon reading the description of the master seal it appears there is a limited amount of them in game. My question is how rare are they actually? Is there enough to promote mostly every unit? Will I run out and then unlock my favorite hero and never be able to promote them?
Yes, from the story mode and history mode combined, there are currently enough Master Seals to upgrade every character that is available, and when the DLC is released, they will include more Master Seals for the new heroes in the new history mode maps.
While the story mode seals are relatively easy to get, the history mode missions require an S ranking on maps that may be difficult, so be sure to diversify the types of heroes that you use the seals on (swords, axes, lances, etc.) so you have a good choice of powered-up warriors when trying to get the others.