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I’m 3L international student student and got a return offer from a biglaw. Though I haven’t signed any contract yet, nothing in the offer letter indicates that I cannot look for other opportunities.

My first lottery draw will be next year but recently chances aren’t really promising. While it’s technically legal to have multiple employers to sponsor H1B lottery for the same person in the same year, I’m wondering whether it is professional to have as many firms as possible to draw my lottery, or whether it will risk my current return offer. Additionally, do I have any chance if I tell firms that I’ll withdraw the offer if it doesn’t get me selected for the visa?

Your question is a little confusing, I think mostly because you may be under a misapprehension. You will not be signing any contract: employment in the U.S. is by-and-large - including in Biglaw - considered "at will." You don't owe your employer anything, and don't be confused that they owe you anything, because they're certainly not thinking that way.

You also say that you might "withdraw the offer"? The firm is the one making the offer; I suppose you mean that if you don't win the visa lottery, you'd have to withdraw your acceptance. That's fine and Biglaw employers understand that - but I see no reason for you to be telling them unless/until they ask. Hopefully that's something you work out with HR well after offer/acceptance.

Without commenting at all on the legality of it, ultimately, it may not be entirely "professional" to have multiple employers trying to secure your visa, but it's probably also a harmless situation. First, you're unlikely to end up in a situation where more than one of the applications actually succeeds. Second, how else would any of them find out that you're working with another potential employer? You have to protect your own interests. There are firms out there who will post you to an international office if the U.S. visa doesn't come through, but many won't - so don't hesitate for a moment to do what you need to do to advance your career.