Aligator clips on those little arms aka "Helping hands". And patience...
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Back in the day when I learned soldering, the master told me that usually I need 3 hands:
One for the iron, one for the soldering wire, one for the piece that I want to fix (while the other piece is already fixated, for example the board)
Before I used the "3rd hand tool" with the alligator clamps, I often held the soldering wire with my mouth and moved it precisely into place. It takes some practising, but it is doable.
So maybe this is a way for you: always use that "3rd hand tool" AND use your mouth for the soldering wire.
This needs to be said less and less these days, but Hopefully lead free solder for the mouth.
There might be tools people have designed on sites like thingiverse to help with this as well.
even fumes from lead free solder make me feel ill and cause headaches if I don't use an extractor and keep my face back. Holding it with my mouth seems like a bit much.
fumes from lead free solder make me feel ill
That's the (doubtful) aroma of the flux that's inside the wire, not the metal itself. You would touch only the metal with your lips.
Word of encouragement, I'm sure it can be done: my great-uncle got his arm and leg blown off in WWII - he founded a family, worked, went to summer holidays abroad in his camper with the family every year - and he also was a stained glass artist who somehow used his one hand to cut small pieces of glass into exact shapes and solder them together with lead. I only saw his setup once and it's 24 years ago - so I can't remember details. I think there were plenty of different types of clamps and a work area with an edge to press things against and keep them from sliding off the table.
I guess take it as a challenge. If it's for hobby stuff then congrats: you now have an even more challenging hobby to get obsessed with. I imagine that you could try to find creators of all kinds of crafts and diy stuff who work with challenges similar to yours to get ideas. I'm pretty sure my late great-uncle is cheering you on from the afterlife, you've got this!
Auto feed soldering gun
A local hackerspace will probably take pity on you and find someone to do this for you.