The tooling to make part-specific press dies is going to be very expensive to set up, especially for stainless steel and I suspect that you would have to modify the design somewhat to make it a viable pressing.
By far the best solution for anything other than pretty large volumes is going to be having them fabricated.
You might be able to find the main body as an off the shelf pressing otherwise just weld a cap onto a length of tube.
The plate on the base could be laser cut and the tubes on top are just lengths of tube, both of whcih could be TIG welded on.
That is all pretty straightforward. The only slight complication is the rounded shoulder at the top of the main cylinder. The simplest solution would be to just simplify the design to a flat plate cap on the end of a tube. If it absolutely needs the radius then you could get it machined or pressed. It should be simple enough not to be too cost prohibitive. For example a simple dished shape for the cap would be vastly easier to tool than a press die for the whole part.
To expand on what I've said in the comments this looks like a fairly straightforward welding job, especially as the volume would certainly justify making a fairly sophisticated jig but you can let the manufacturer worry about the fine detail.
the only possible issue I can see is that it is a bit tight for space in the gap between the two small tubes it get in to weld it but that doesn't look like a major issue.