Greenhouse wood foundation and frost heave
The short answer is: No, don't worry about it.

Your building is basically a yard barn and no building code I have seen requires frost footings for a yard barn.

I live in a Zone 5 with plenty of freezing and snow and separate buildings like garages are only required to have a 12" slab on grade footing. Yard barns can be place right on the grass if you want. Although, most people scape off the topsoil and place gravel to level the barn and provide some moisture reduction.

I don't think your building is large enough to be affected by frost heave in any appreciable way.

All that said, if you are an ovebuilder and want this to be permanent then a deep sonotube footing in each corner would certainly keep it from going anywhere. Anchoring it to these footings would also give you some wind stability.

Good luck with your project!