Can I replace Shimano 105 5500 shifters and derailers with new Sora R3000 ones? I have a 2000 Klein Quantum
(this one, 2x9 speed: https://www.bicyclebluebook.com/value-guide/product/45657/)
My shifters have been replaced once already with used 105 (also 5500), but now they have a lot of issues and I would slowly like to upgrade the whole groupset.
Can I start by switching out the shifters and derailers with new Sora R3000 parts (also 9 speed)? Or would I have issues with the brakes/crankset/etc?
This would put me back on the road while I save for the rest.
Thanks for any help, this is my first time here.
Can I start by switching out the shifters and derailers with new Sora R3000 parts (also 9 speed)?

All Shimano 9-speed parts use the same derailleur pull ratio (except the front derailleur pull ratio can vary between MTB / road components). Thus, you don't even need to change the derailleur, just changing the shifters is enough.
The STI levers are short-pull levers compatible with any cantilever or caliper brake. They are not compatible with V-brakes. As your old STI shifter/levers were short-pull too, no changes there.
Just be sure to select the Sora 2x9 parts instead of going for Sora 3x9 parts.
I'm not sure what issues you have but as the STI levers are expensive, please be sure to eliminate the possibility of issues coming from some other parts first. Perhaps your bike needs new inner and outer cables instead of needing new shifters?