I live within 5 miles of a small private airport, how do I find a phone number to call?

Just stop by.

You can look up the number of the airport and give them a call asking them if it is okay. If they do not pickup, you can always drive to it and ask.

Go to: https://skyvector.com/
Open the tab Airports, open the tab for your country, then state, then the airport.
Look for Airport Manager. There will be a phone number there.
By the way, here in the US the airspace around private airports is normally class E, so you don't need anybody's permission to fly in it, but to call and ask about the traffic pattern is always good.

I like airmap.io.  

https://app.airmap.io/

It should start up at your location.  Click Recreational or Commercial at the upper right.  You should see the circle around the nearest airport.  Clicking within the circle should bring up the phone number.  

As a hobbyist, you're required by law to notify the airport of your flight plan in advance if you're flying within five miles.  If you fly often within the circle, you can probably set up some kind of verbal agreement.  Or you can agree to just text them which is what I do.  

Thank you very much. that helped a lot.

Mark The Droner Posted at 2017-3-8 04:19
I like airmap.io.  

https://app.airmap.io/

"As a hobbyist, you're required by law to notify the airport of your flight plan in advance if you're flying within five miles."

Mark, where did you get this information from?
Can you give us a direct reference?

piowoc73 Posted at 2017-3-8 10:56
"As a hobbyist, you're required by law to notify the airport of your flight plan in advance if you're flying within five miles."

Mark, where did you get this information from?

In the US, it's public law 112-95 sec 336 and also FAR Part 101.  

Sorry - been off the grid for a while

Attach a sign to your drone ie "Is this okay", fly by the tower and they will find you.