Does the EASA LAPL(A) have a FAA equivalent?
I'm not sure about their equivalence (meaning one cannot fly in US with a LAPL right way), but the closest that comes is the Recreational Pilot license. 

The requirements are roughly similar:


30 hours flying experience; however, the FAA requires 3 hours of solo flying and cross country requirements are slightly different.


The limitations are also roughly similar:


Recreational pilot license doesn't allow the pilot to have more than one passenger, while the LAPL does (upto three).