Hi Kevin, I'd use this - Remington 700 Varmint SF .223 budsgunshop.com/catalog/product_info.php/products_id/56458 If there's a classic prairie dog rifle, this is it. I've had mine for close to 20 years, and the design hasn't changed much. It doesn't need to.
The 700 is legendary for out-of-the-box accuracy, and with a 26-inch bull barrel, this rifle milks all the performance possible from the design. Because sometimes you just want to reach out to a prairie dog so far away that he chirps with a goofy accent.

It depends on what type of hunting you want to pursue. If it's normal shooting out to 300 yards any heavy barrel bolt gun in .223/5.56 will do and most factory 55 grain ammo will do. If you want to stretch out to 500 yards or more my Savage 11 Long Range Hunter in 6.5 Creedmoor with the Vortex scope really shines. Winchester's 140 grain match ammo (or my handloads with Hornady's 140 grain SST) has the accuracy to hit them out to 700 yards so far.

Prairie dogs have very little paws and I’m not sure you’ll be able to find a rifle small enough for it’s little paw.

I would recommend a Savage 19174 AXIS XP 223 Stainless W/SCOPE. budsgunshop.com/catalog/mobile/product/70982/savage_19174_axis_xp_223_stainless_wscope - This will be a hard to beat, accurate and all inclusive prairie dog package for the money spent.

Heres a few more models. outdoorlife.com/photos/gallery/hunting/2011/06/best-rifles-hunting-prairie-dogs

cal..223 I like Ruger and CZ budsgunshop.com/catalog/product_info.php/cPath/2084_36_59_110/products_id/61806/Ruger+17875+Hawkeye+Varmint+Target+4%2B1+223REM+26 **********************
budsgunshop.com/catalog/product_info.php/cPath/2084_36_63_971/products_id/97563/CZ-USA+CZ527+American+223+1%3A9+WALNUT