If I have an entire font in images or vector files, is it possible to use Illustrator to create a font file that can be used on other systems?
What Adobe product do I need to use to create a .ttf file if I cannot use Illustrator?
If your needs are pretty straight forward, I second the Fontself recommendation. Inexpensive with free updates.
Ideally, you'd use dedicated font editing software for that.
If you search "font export plugin for Illustrator" you may find something that suits your needs.
If you want to use images (pixels) in your font, you will probably have to go into the font software specifications if they can do that.
There is a version of Fontself for Photoshop which can do it.