Community college! The PCC[1] welding program costs $5200 for a 1-year certificate. They also offer training in a variety of other trades for roughly the same price.
IvyTech is really the only 'community college' and in fact is 2 of the 3 schools that come up when you search 'Indianapolis community college' (the other I've never heard of and given the neighborhood it is in I doubt it's a proper educational program) in Google. It is $140 a credit hour (which isn't true, they add tons of fees) for in-state students. Using your example the only welding certificate they have is Structural Welding at 21 credit hours which after all fees appears to be $4690 for in-state students. It appears only 20% of their students actually complete the program on time (within 8 months) too which doesn't say much for the school... https://www.ivytech.edu/academics/gainfulemployment/ge/struc...
IUPUI, hands down the cheapest accredited university in Indianapolis, doesn't have a 'welding certificate' or anything remotely approximating it. They do have 'certificates' on their minors page but the only thing remotely trade related is some certificates in motorsports
[1] https://www.pcc.edu/programs/welding/