Yes, you can sign a contract to sell property or rights you do not yet own. Outside the tax sales area, this is common with options, or even just flipping real estate purchase contracts.
You need to have an "out" in case you are not able to purchase the tax rights within a certain time period. It does not have to be done through a closing attorney, but an attorney will have to prepare the deed. Usually there is a $50 to $75 charge for that. If the buyer or the seller is an individual (not an LLC or similar) the party that is an individual can prepare the deed. In Alabama, the deed must say the name and address of the preparer. In Tuscaloosa county, the deed must include the Source of Title. Maybe other counties are similar. I know Jefferson does not require it. There is also a required from that includes information about the transaction. Look up almost any deed filed after 2019 or so, and you'll see the form. You can include the same information in the body of the deed, but most people use the form. If the seller is married or single, you have to say that. If it is a local or out of state entity, you have to say that. (Example: Alpha LLC, an Alabama limited liability company or Bravo, Inc., a Delaware corporation)
Buying over the counter from Alabama Department of Revenue can take a long time. Once you receive a quote, you have 10 days to complete your purchase. Because of wholesaler activity, and people asking for quotes on 5 - 10 different days in a row (extending the effective time period to 50 to 100 days) there is a tremendous backlog in some counties. I have heard of people interested in Jefferson County properties being told they will not get a quote for 4 or 5 years because of that.
Redemption rights in Alabama are exceptionally complicated. Not that difficult once an expert explains them, but it takes a while to explain. And, you have to be able to remember a large number of moving parts to know the answer in any particular situation. There are a lot of so-called experts spouting stuff that is completely wrong because they never completely understood it the first time somebody knowledgeable explained it to them. Or, they never talked to anyone knowledgeable, but just picked up advice for investors who have been lucky, not educated.
Do your homework!