If you have a steamer it would help.
Else diluted degrease .
To play safe, can go for fabric cleaner.

Tools you need : vacuum(wet dry vacuum preferred), steamer(or hot pail of water), microfiber clothes, brush to agitate

1.Vacuum and remove as much Gunk's or vomit as possible
2. Choice of cleaner : APC/diluted degreaser/fabric cleaner
3. Steam or wipe with hot microfiber towel
4. More cleaner and agitate with brush
5. Remove as much soap from seat as much as possible with dry microfiber clothes
6. Repeat if necessary
7. Air dry/ or pet dryer if you have it

No steamer can always opt for iron, of course the kind that produces steam.
just don't steam too long.
No choice but to make few passed. Vomits are very acidic. So it wouldn't be an easy one.

If you want to go more hardcore, sprinkle baking soda. do it dry. let sit for an hour and vacuum it. You will have to clean it later. it also neutralises acids. SO it's good.

if you have silica packs, just place it on top of your seats to remove moisture.

Yes, fabric cleaner are usually mild, formulated not to discolor fabrics. you can use it on plastic/vinyl too.
Pg quick detailer can be used like bsd too. just saying i've experimented with it and it actually last up to 2 weeks.
It streaks a lot, so best method is to dry car, use a damp microfiber towel to spread and another to buff. Like how you would apply carpro reload.
Currently still using it on my family car and it's good. gives very good shine. it's super economical too.

Guys, for fabric seats, how best to clean it DIY? My car fabric has been abused, with years of neglect. So now the color has become dull, and got faint funky smell.

Though febreeze may handle the smell, it is more like masking the smell with its scent, rather than removing it?

If not investing on any machines, then u need APC, some brushes, give it a good scrub, then microfiber towel to try absorb and dry off the seat. It may require 1-2 days of drying under the sun also.

For the smell, febreeze may have some effect, else u could also try vinegar in spray bottle. Again, it will require some sun to dry it off, but from the theory the acid in vinegar should kill the odour causing bacterias.

If u wanna invest in a machine, then a steam cleaner should be able to both clean and remove the smell.

Alternatively if totally lazy to DIY, then send to car wash shop and get them to do interior cleaning plus nano mist.

Sincere apologies, think u may have misunderstood my reply regarding drying under the sun. I did not mean u had to take your seats out of your car to do the cleaning I mentioned. Instead, once u have done your cleaning of the seats, they may still be damp, so park your car under the hot sun and head to dry them out that way. No seat removal required.

APC is All purpose cleaner, a common cleaning solution that is usually concentrated and require some dilution, good for general cleaning most interior surfaces. There are few different brands in the market for this. Alternatively, a cheap and easily accessible alternative would be either dish soap or laundry detergent. However do use with care as they are not designed specifically for car use.

can use STP Tuff Stuff.