Not sure if this us a pun, necessarily, but here goes.
A termite walks into a bar and asks " Is the bartender here"?

What did the lady at the beach say to the Catholic Priest?
Excuse me, sir, you're in my sun.

Abstinence makes the church grow fondlers

I sent 10 puns to the local radio programme to see if any of them would get broadcast but no pun in 10 did!

LEGO my Eggo

It’s hard to explain puns to kleptomaniacs because they always take things literally

Atheism is a non- prophet organization

I hate puns but I offer this: two nuts went walking in a park, one was a salted. (ugh in advance)

What do you call a man with no arms and no legs...

...floating in a lake? (Bob)
...laying on the floor? (Matt)
...hanging on a wall? (Art)
...(two of them) in a window? (Curt 'n Rod)

Blackjack dealer in Deadwood rapid-fired them at us--had me in stitches. Can't remember them all.

Not all are puns, but close:

I've invented a new word. It's "plagiarism".
I have the heart of a lion and a lifetime ban from the Toronto zoo.
How do you think the unthinkable? .... With an itheberg.
An old lady at the bank asked me to help her check her balance. So I pushed her over.
My friend says to me, "What rhymes with orange?" I said, "No it doesn't."
Puns are bad, but poetry is verse.

For real