I have road bike with old 105 5500 2x9 groupset. My crankset chainrings worn out from wear and need to be changed. My crankset is 105 FC-5501 52/39 Octalink 130 BCD. I looked for 5500 series chainrings in e-shops, but I didn't find anything for this old model except e-bay where I can get all crankset with crank arms. Could you advise which ones of chainrings would be appropriate with FC-5501? Can I use for example a Shimano 105 5700 10 speed 130 BCD chainrings on my crankset?
Shimano might not say it's okay, but you can use any 130BCD chainrings that you can find with that crankset1. You'll get the best shifting performance if you pick a matched set of chainrings from a single vendor. The one thing to avoid is any chainrings that say Narrow/wide in the description, these are 1x specific and won't work well on a double crankset.
When replacing chainrings, it's important to line them up with the crankarm for best shifting. There will be a mark or bump in the chainring that indicates where the chainring should be installed relative to the crank arm. The outer chainring generally has a small metal pin to prevent the chain getting wedged in the crank arm in addition to this bump. The whole process will be much easier if you get a chainring nut wrench like the Park CNW_2.
If you've ridden the bike long enough to need new chainrings, you will likely need to replace the chain and cassette as well. Chainrings are generally the most durable of the 3 drivetrain components.
While you are buying stuff, get a chain checker. Replacing your chain before it wears will greatly increase the life of your chainrings and cassette.
1- I know that everything up to 11 spd will work fine, I've no idea about 12 spd stuff as I have no experience with that yet.