First of all, as mentioned above, I suggest you ascertain if CO will get you into SXM rather than SJU.

Unfortunately, it appears that WinAir no longer operates scheduled Twin Otter service between SXM and AXA (and it also looks like AA Eagle no longer operates ATR-72 service SJU-AXA as well).

However, there's hope: it's a fairly short taxi ride from SXM across the island to the marine ferry terminal on the north (French) side of the island. We've done this several times and it's no problem.

BTW, Anguilla is a wonderful place. Check out Johnno's Beach Stop (restaurant and bar right on the beach, natch) and inquire about his day cruises to Prickly Pear island via powerboat. And also be sure to check out the Sunset Beach bar right next to the end of the runway at SXM if you can change your travel plans and fly into this interesting airfield instead of SJU.