It is ambiguous. It needs a comma to distinguish between:
“Pass either exam 480, or 483 along with exam 486”
= 480 or (483 +486)
and
“Pass either exam 480 or 483, along with exam 486”
= (480 or 483) + 486
The only argument for the one option is the lack of qualification of 483 by “exam”, favouring the second. But I regard that as weak at best, certainly not dispelling the ambiguity. The comma, I believe, does.
Additional observation
I find “along with” poor. “Together with” might be better, but “and” has exactly the same meaning and two fewer syllables. This is does not address the ambiguity, but, all things being equal, fewer words give greater clarity.