Is there a five or six speed automatic transmission that is compatible for a 98 base Camaro (3.8L V6 engine with four speed automatic transmission)?

 

You can make anything work but you are talking about some custom hardware and and a custom Stand alone controller. If you are really into you could probably do a megasquirt system to run everything but by the time you are done it will cost you more then the car is worth. Keep in mind torque and gearing are a big part of the transmission design. There has been one guy that swapped his T5 5 speed for a T56 6 speed, This was a costly swap and in the end he could not use 6 gear much at all. The motor just did not have the torque to use 6 gear under 2000 and it did not have the HP to have a higher top end. So both is top end and MPG dropped slightly. The only real gain he got was a much stronger transmission he could beat the crap out of.

FYI I think the trans you are looking for is 6l50e if you really want to try. That tech may be old enough now to just get a stand alone controller and you could get a PCM out of a 98 with a manual transmission. Things like making the speedo work are going to be a pain.

 

The short answer is no.
Is your trans bad? If so get it rebuilt with a shift kit 4 speed is all you need.