I just bought a JR XP783 transmitter and I have a reciever but I just want to know if I need a matching crystal in the reciever or if I can fiddle with it a little bit to make them work without having to buy a whole new reciever.

I do not know what kind of receiver you already have but these are a few things to consider:
If you have a JR brand receiver, then the frecuency crystal should match the channel used on your radio (In example Transmitter is on channel 49, receiver crystal should be the same.
If you have other receiver brand you have to make sure that it works in positive frecuency and also check that the channel match.
All futaba receivers are negative frecuencies and will not work with your JR transmitter, not even if they are in the same channel.
Some Hitec receivers are made with frecuency shift so they can work with both systems JR and Futaba, some receivers have a plug that must be turned to work on positive or negative.

I hope this can be of some help.

I forgot to mention that almost all receivers have an interchangeable crystal, so you do not need to buy a new receiver but just the matching crystal.

You need a crystal on the same channel as your transmitter. With that, it'll work. Without it, it won't. You can change the crystal in your receiver.