The old Bias Ply tires were really bad about developing Flat Spots while parked overnight. Felt like the wheels were square till the flats softened up and the tires got round again.
Radials, usually not so much. The tires that acted square for me were on a one-ton van. It'd sit in the garage like the wheels were blocked. Put radials on it, I could then push it out of the garage easily by hand.
What kind of tires are on this, how old are they (date code after the DOT codes, usually on only one place on each tire) and what is the tire pressure?
It's possible there's a stuck brake caliper. Those'll shake your wheel, make you think you're driving on a flat. Drive awhile, pull over, and see if one front brake is noticeably hotter than the other. When that happened to me, the left wasn't anything unusual, but the right was like walking up to a burning charcoal BBQ.
Date Code on Tires: Last four digits. First two are Week of the year, so 01-52. Second two are Yearm so 0414 would mean made in the fourth week (January) of 2014. Older than five years, tires need replaced, even if they look good.
Brake work isn't outrageously expensive, so if the shake is tires or brakes, not likely a deal breaker.