I know that it is difficult to answer these questions without knowing the full scheme it would be used in, but are there any problems that could be introduced into a block cipher if an involutive S-box is used?
As pointed out by Aleph and poncho, it is difficult to provide an educated answer without seeing the whole cipher.
One one hand, involutive S-boxes can be used to attack a scheme: e.g., this key-recovery attack against MANTIS observes that since the SBox is involutive, its DDT is symmetric, and uses it against the scheme. This paper by Biryukov shows that involutional components can be used as a leverage to ease distinguishing attacks.
On the other hand, there are block ciphers like ICEBERG which heavily rely on involutional components and are still secure: in ICEBERG, the SBox is involutional, but also the multiplication matrix and even the whole cipher itself (up to the key schedule).
Also keep in mind that the rationale usually given for putting involutional components in block ciphers is that they allow compact hardware implementations (you can reuse the same components for encryption and decryption). So if hardware implementation is not your goal, I would advise against using involutions as they can introduce subtle weaknesses.* As there already exist many great block ciphers, I would also strongly advise against designing your own.*
*unless there is a strong incentive to do so (or if you are doing it for research/recreational purposes)