Going to build a pedestal desk with a 2x2 frame with joints like these:

http://ana-white.com/sites/default/files/modern%20desk%20base%20plans%20easy%20step%201.jpg

I'm looking at the Kreg R3 Jr jig for doing the pocket holes. Anyone know if that will work for 2x2 material or if it would be too wide for wood that narrow?

R3 is rated 1/2 to 1-1/2 wood.
I would be very cautious on the small side.
Are you using 2x dimension lumber (1-1/2) or actual 2 inch net lumber?

2x2 stock so 1.5"

Better mock it up first.
It is liable to look a little spindly with smaller sections.
Glue and screw the joints.
Franklin Liquid Hide Glue will make very rigid joints.
Coat end grain and allow it to soak in and set up.  Then apply more glue and let the pocket screws clamp the joint by pulling it tight.

If you are going to paint it consider getting some actual poplar in 8/4.
That should come out around 1-3/4 to 1-7/8
If you can find 10/4 you could easily get a full 2 inches net.

Laminating up from thinner works well under solid finishes not as well under clear finishes.

Hardwoods get pricey sort of fast (poplar is not so bad and ash is cheap right now) so using them for utilitarian stuff is often not worth it.

This is the design, desktop and drawers removed to show frame.

What material are you using because that is a big part of it? If you are using a tight grain hardwood then it will probably hold but I wouldn't trust it with softwood.

Hopefully oak or poplar depending on what I can find around here.

you can probably get both oak and poplar 2x2s in the stairway railing section

Why are you making that out of 2x2 wood?  3/4" wood would be just fine for that.  

And you can get red oak and poplar at Lowes or Home Depot.  Kind of blah wood for a desk.  

Stan Houston Equip Co
501 S Marion Rd
Sioux Falls, SD 57106
605-336-3727

Above is a Rockler reseller in SD.  They might have nicer lumber on hand.  You can also mail order wood from a wide variety of sources.  If you go to Lowes or Home Depot, go to the magazine section and pick up a copy of Wood Magazine or Poplar Woodworking.  There are lots of advertisers in both rags that will send you what ever you want.

And the HD hardwood is painfully expensive.

Find a hardwood dealer or even order some from a place like Steve Wall lumber http://www.walllumber.com/
They skim boards to make a bundle that UPS will handle.