Can you play 2 players on FIFA 99?, not via modem, but against someone with the same keyboard, except each person uses different keys.

No, But if you had a Gravis GamePad(Or other game controller), you could have one play with the GamePAD, and the other with the Keyboard.

Normally I wouldn't reply to someone who calls himself ajakkes fan, but in this case I will because the answer of Joe is not complete:

It is possible to play with up to four people at the same time on one computer if you have the gravis GrIP system. You can all use a gamepad as well then! I use it for almost four years now and it is supported already from FIFA 96.