I transferred some bitcoin to my bitcoin core wallet, at the bottom it is still showing "Synchronizing with network" and my funds have not yet appeared.
That's normal. If your transaction is included in block 100, then you won't see that transaction until your Bitcoin client downloads block 100. That means that you won't see it until your client is almost done with synchronization.
Did transferring funds to my wallet before this process was complete cause my funds to not go through?
You can send Bitcoin to an address made by a client that is not done synchronizing. Your client cannot tell you what your balance is, but the balance exists anyway. Once synchronized, the client will show the money, if there is any.
The site I transferred from did say successful transaction.
That doesn't necessarily mean that the transaction was successful, or that the transaction exists at all. They can say whatever they want.