You may need to clean out the fuel tanks, petcocks, and strainer screens on those Ranchers. If they sit with fuel for a long period of time they can get pretty rotten. Ethanol-laced fuel is especially bad... Sometimes the strainer screen degrades and needs replacement and sometimes the petcock valve corrodes and clogs up. All of those parts are available new. Don't buy aftermarket (china garbage) parts, OEM Honda are the only parts that will work.
https://www.partzilla.com/catalog/honda/atv/2003
You can test the fuel flow by turning the petcock off, take the fuel hose off from the carb and aim the end into a bottle. Turn the petcock valve on and note the fuel volume flowing into the bottle. If it flows very slowly loosen the fuel cap and see if that helps. If so, the fuel cap vent hose might be plugged up. If fuel does not flow good after loosening the fuel cap everything in the tank will need to be inspected, cleaned or replaced as explained above.
Welcome to the forums!
https://www.partzilla.com/catalog/honda/atv/2003
You can test the fuel flow by turning the petcock off, take the fuel hose off from the carb and aim the end into a bottle. Turn the petcock valve on and note the fuel volume flowing into the bottle. If it flows very slowly loosen the fuel cap and see if that helps. If so, the fuel cap vent hose might be plugged up. If fuel does not flow good after loosening the fuel cap everything in the tank will need to be inspected, cleaned or replaced as explained above.
Welcome to the forums!