"stop hounding us" would mean "stop bothering us" or "stop coming after us (because you want something)".
"Hounds" are dogs that are used in hunting:
Because they are for that purpose, they are not supposed to give up. They keep going after the prey (even though the prey doesn't want them to). So to "hound" is to follow or pester or bother someone about something.
A similar phrase would be "stop badgering us", since badgers are known for being very persistent in a fight, and do not give up easily.
"stop hounding us for puns"
A pun is a kind of joke based on wordplay:
So we'd assume these people have made such jokes in the past. Now that they have done so, everyone wants them to make more--when they want to be more serious or get work done. They are asking to stop bothering them to make more of these jokes (apparently without actually being bothered).