If it comes with the top, then it will not work. If you're adding your own top, then you have it made to fit. You will need fillers on the sides of the vanity.

Thanks for responses. If I order vanity with top that is 36 inches, will buying side splash be a solution to filling the gaps?

You don’t order a vanity with a pre made top in your situation. You order a custom top from a local fabricator. Pre made tops are for freestanding vanities without walls in each side. Not built ins. And this is where most people buying online go off the rails. They don’t understand the difference, and order the wrong things. Buying online ends up costing you way more in frustration and money both. Just make an appointment with a kitchen and bath place.

You have two problems, the countertop and the vanity itself.

The countertop is 36" wide but the opening is 37.5" wide. That leaves a 3/4" space on each side. You can't just use side splashes to cover the gaps;. side splashes sit on the countertop and you won't have any countertop for them to sit on. The solution is to buy a vanity without a countertop. You can then hire someone to make a countertop for you. He will make a template of the space and cut a countertop to match. By templating the space, the countertop will fit perfectly even if the walls are not square.

Next, you'll have 3/4" between each the side of the vanity and the wall. Perhaps you can live with the gaps. If not, you need to find material that matches the type used on the front of the vanity and cut it to fill the gaps. I had gaps between the walls and the cabinets I installed in my IKEA kitchen. Fortunately, IKEA sold matching "decor strips" that I cut to the required width on a table saw and glued in place to fill the gaps.*

*If you couldn't find material that matches, you could use something that contrasts, even metal trim.

-- amateur

Any chance backer rod would work and look ok? I'd like for the countertop to be caulked and flush with the side walls. I'm okay with leaving the vanity cabinet sides with a gap, as long as I can get the top caulked to the walls.

No. Backer rod will look like a big mistake. Because it is. If you want to use a free standing vsnity in an alcove, you have to go smaller. 24” in your 37.5” space. Otherwise, you do a built in, and do the fillers for the vanity, and the custom top.

I used backer rod. The vanity I bought had a 4" high acrylic sink slightly wider than the vanity itself and that flared out slightly from top to bottom. I shoved backer rod between the side of the sink and the wall, placed aluminum trim over it and used adhesive caulk to secure the trim. The trim finish matches the finish on the vanity door and drawer handles.