I need to install new SFP's in a location; however, I need them to connect to existing switches and I'm not sure what SFP type they are. All I know is I've got MM Fiber.
If I installed Cisco GLC-SX-MM 1000Base SFP's, would that cause an issue if the other end of that fiber was connected to a 100Base SFP, or do I need to have matching speeds on both ends?
Your speeds must match
There are some handy links in this other serverfault question which might help.
How compatible are sfp's really?
It depends on what you mean by cause an issue. It won't break anything but you won't get a link either. They must be the same. SFPs are hot-swappable so you can remove it to inspect the label without damaging it. Additionally, you can do a show interface status to have the switch tell you the model SFP that is installed in a port.
Not only can you not mix speeds but you should stay away from mixing sfp models as well.