Really, a ,local Briggs dealer will have what you need and you won't pay shipping. Typically these aren't serviced as a kit I don't believe. The float valve comes in a kit, the bowl seals either come separate or with the bowl complete , gaskets are on their own etc.

After doing some checking, the idea is not to even think about a parts kit - just get a new carb from the B&S dealer. How does the gas tank come off? I want to be sure that's clean before I put a new carb on the unit.

Kudos to you for thinking to pull the tank! Once the plastic cover is off the top of the engine, there are 3) 5/16" hex head cap screws and one 3/8" hex head bolt on the side below the fuel cap. A spacer is located between the block and tank on this bolt- so watch for it. You will want to flush or replace the fuel line too.It may be a good time to remove the shroud and clean the cooling system if you are so inclined.
Back to the carb. These really are the easiest carbs to clean and unless it has been damaged w/ water- they clean up successfully. When I service these, 9 out of 10 don't get removed. I clean the bowl nut and bowl without ever removing the airbox. If it has sat for a while or has float valve issues, then of it comes.

Looks like to me that the bowl nut/bowl sit so tight up against the frame that one must take the air filter stuff off before bowl/nut removal. This mower - not mine - has been sitting out in the elements for "who knows how long " and is in really tough shape. One thing I checked first was compression. It has that and it does start with starter fluid - so that's a plus.
Thanks for your ideas.

As tomplum stated these are the easiest carbs to service. I can not remember the last time I needed to buy any parts for these carbs. One other thing I found replacing the fuel line is the best way of preventing the carb from filling-up with bits of rubber.