Is this serious?

I don't know if this vehicle is on an E series chassis but if so, I have been told multiple times that they are prone to ball joint problems. You might include that in your list of things to check. Those that say it might be the tires are probably right but, something caused the tire problem. My previous E series Dynamax had this issue. It was a year or two newer. I bought it new and I only had about the same relatively low miles when it showed up. Best of luck trouble shooting.

Are you sure the transmission was shifting correctly through all the gears? Could it be trans shudder not shifting?

Be aware that date codes for 1999 and earlier are only three digits, the last being the year, the first two are the week of the year. If the tires are original (highly possible) then the date codes may be three digit for 1999 tires.

http://www.tirerack.com/tires/tiretech/techpage.jsp?techid=11

Date codes are only found on one side of the tire, so you may have to look on the inside sidewall of some tires to find it, depending on how they were mounted.

Charles

Get the suspension checked out, for sure. But it sounds more like a tire issue.

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.

I had a tire that would bounce at 10 ~ 5 mph or less when slowing down. Wouldn't have thought it was possible but a simple balancing fixed it.
For the few dollars it would cost, I would have the tire balanced and see if that stops it.
Otherwise it may be much more expensive.

What did the tire shop tell you when you stopped there and had them test drive it?