In most states an original SSN card isn't a specific requirement. It could be something like a 1099 or a W-2 with the SSN listed. If you have anything like that with your full name it should work if you're really worried.
However, if you're really worried about it you can request a new SSN card with a complete name. I don't believe they've used middle initials in a while on SSN cards.
I found this from the Pennsylvania DMV:
Middle Names vs. Middle Initials
Another common scenario is that a proof of identity document (birth certificate or passport) will show a person's full legal name, but subsequent documents (social security card, proof of residency) will only show a middle initial. As long as the middle initial on the subsequent documents is the same as the first initial of your middle name as reflected on your proof of identity document, it is acceptable for REAL ID.
For example, if your birth certificate shows "John James Doe" and your social security card and proofs of residency show "John J. Doe," this is acceptable for the purposes of REAL ID verification.
However, REAL ID regulations require that your full legal name (not middle initial) is shown on your REAL ID driver's license or photo ID card. If your current driver's license or photo ID only includes your middle initial, PennDOT staff will be required to process a name change so that your full legal name is reflected on your REAL ID product.