Can I do an IFR checkride in an airplane that isn't certified for IMC?
Legally, you can take your instrument rating checkride (not "IFR checkride") in an airplane that is not equipped for flight under instrument flight rules. That is because you will fly your check ride under VFR. As I did. The examiner will act as "safety pilot," a require crew member, while you are the sole manipulator of the controls under SIMULATED IMC via the view limiting device. The examiner will ask you to do the required tasks, and you will coordinate the tasks needing ATC assistance with ATC, who will remind you to "remain VFR" while you are on practice instrument approaches and so on. You cannot fly the checkride under instrument flight rules because you don't have an instrument rating. The examiner won't (or at least most won't) file an IFR flight plan as PIC using his/her rating.

But that's legal. In practical terms, at least in my opinion, any examiner is going to be reluctant, or at least suspicious, of a request for an applicant to complete the instrument rating checkride in an airplane not fully equipped for instrument flight. In particular, he or she is going to insist that the airplane be equipped for several different types of instrument approaches. My understanding is that it's up to the examiner, so check in advance if your examiner is comfortable using the airplane that you propose to use.