We were wondering if we could have custom music made by an outside source provided we credit them.

Great question!

Yes.
For the game competition they've recently revised the rules as follows:

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Rules Addendum II: Non-Team Asset Work Contrubution

If other students who are not directly on the team want to contribute an asset, but do not want to contribute on the team itself..

• Out of team assets from students are allowed as long as the total non-direct team member additions to the project are less than 10% of the total project itself. This includes all aspects of the game, and everyone who has worked on it must be credited, along with their contribution. 
  

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And we're honoring that rules change in the film/animation competition  as well (IIRC rules haven't been formally updated yet, but consider this formal notification).

Just want to re-emphasize the need to cite/credit (which you definitely said as well, but in case other people are reading this too!). External music/etc *WITHOUT* citation is potentially grounds for disqualification.