I am doing research for this answer from time to time and this answer is going to be updated as I learn new things. Feel free to suggest edits or post your own answer if you have better information.
There are five problems that need to be solved when looking for a replacement rubber band:
This is probably the easiest. There are a lot of rubber bands available in every corner of the Internet and for most cases you'll likely be okay with slight variation in size.
Height being how tall the band is when you lie it down - too tall and it won't fit on bushes or wedge belts. Sometimes seller will specify this dimension, if not you can always try asking.
For most intents and purposes thickness should not matter much, unless you're dealing with severely limited space which in my experience is rarely an issue with official sets.
LEGO rubber bands come in two cross-sections - square/rectangle and circle. While I've found countless square/rectangle rubber bands I'm yet to find a single seller who deals with round. Even on Aliexpress which is said to have everything available.
Unless you're lucky you'll have to manage with square ones.
Tensile strenght & stretchability
LEGO sets rarely deal with forces sufficient to snap a rubber band, though how much energy is required to stretch it might be important. That's not an easily found information, so you have to deal with it by either contact the seller (or even better manufacturer), or buying different kinds and trying them out.
Even though the question is about finding alternatives, your first line of action should be seeing if the official ones are available.
- Check on Bricklink and other LEGO marketplaces if the specific rubber band you're looking for is available in an acceptable price. Just remember that rubber bands from very old sets even in mint condition can be weakened by old age.
- There is an official rubber band pack from LEGO (2000707-1) which can be fairly easily found but it's not cheap. There are likely other sets. Make sure to email LEGO customer support requesting they release new sets of rubber bands in different colors - the more emails they get the more likely they'll actually do something about it.
If the above are not acceptable options here is what I'd do:
- Assemble the set that requires the rubber bands as much as you can. Use any bands you have at home to finish it.
- Determine what's the acceptable difference in size, strength, stretchability and color.
- Repeat the same process for all other sets you own that are missing bands.
- Shop online for bands that are the closest to your requirements. Amazon, EBay, local equivalents. Maybe you can find a local manufacturer or distributor - don't hesitate to contact them.