My lawn mower and wheeled weed whacker will only run if I am pumping the primer. How can I get them to run constantly without pushing the primer. They are both made by YTD

Probably the carb needs to be re-built. I'd check the air cleaner while your at it.

How do I rebuild the carburator? Is this something I can do? How much would it cost for a lawn mower repair person to fix?

If it's the carb, you can always take off the carb, get a can of carb cleaner, and clear out any carbon and soot with the carb cleaner. That's only if it's due to buildup. If the carb isn't grimed up and it's doing what you are describing, then it may be time for a carb rebuild kit. I'd personally start with a good tuneup. Start by checking the air cleaner, check the spark plug to see if it needs to be replaced (or if the gap is correct), and then empty the gas tank and see if there is anything clogging the gas line or crap in the tank. It's a lot easier (and cheaper) to check and replace the standard maintenance items, then it is to start repairing or replacing larger cost parts that might not need it.

What brand of engines are we discussing?
1. probably old stale fuel, or fuel with water in it.
2. could be trash in the tank or carb.
3. have you been doing regular maintenance in the past?