What kind of table or stand do y'all use for a fly press? 

Ideally I want something i can move when i need to. I have a small shop and I'm trying to use the space efficiently 

Mine came with the press and is fairly heavy cast iron.  I still bolted it down to the slab.  An immobile fly press support system of some kind is an essential part of having your press perform correctly in my opinion.  If nothing else I would go with removable bolts down to lags in the floor.

Unfortunately mine didn't come with a stand. Is there any companies that still make a stand? I'm thinking I could maybe get a large log section from one of the tree service companies and use that. I'm not sure how I'd get the ends flat and parallel though 

I'm not aware of any company making stands specifically for fly presses, and I don't think the people who import new presses from India also import bases.

That said, as Latticino notes, you need the press to be as immovable as possible. Mine is much the same size as his, and while I'm not able to bolt it to the floor, I did add some shelves underneath to hold tooling and add weight. It helps somewhat.

I also made a removable set of casters that allow me to move my press around the shop as needed. Here's a video of how it works:

 

Can you weld? this is totally solid and silent. I would second that it needs to be bolted down really.