As preface, if you haven't been to the Craspire website, they have acres of stamps for very cheap, and I think the vast majority of sellers on Amazon/Ebay/Etsy actually just dropship from Craspire. If you're looking for variety, I'd definitely look there. (If you have, again, apologies, it just took me a while to find it and I spent more than I should have on a couple of my stamps before I started buying direct from them, so I like to make sure I mention it.)

There are a ton of shops on Etsy offering custom stamps, with wildly varying prices -- when I was doing research I saw one that was $55 for just the stamp, and another that was $30 because you couldn't buy the stamp without a starter-kit bundled in (furnace, spoon, wax). Definitely don't pay that much. I didn't get super exhaustive with my research because I couldn't really find a better price than a shop that I'd already done business with, and I knew they were reliable, so I just stuck with them. The shop is PraeyDesigns -- they have reasonable rates overall and everything I've bought from them has been good quality, so they're one of my go-tos for both stamps and wax (there's cheaper wax available but I like their selection). They offer a number of different sales pages for custom seals, so I'll link direct to this one, the one I purchased.

On that page, the third image lists the sizes they offer -- I bought the 3cm oval because I needed it "long" to accommodate the tail on my image. For reference, most larger stamps you buy prefab are 4cm rounds, most of the smaller stamps are 3cm rounds; that particular page only offers up to 3cm but PraeyDesigns does have other custom stamp pages that offer 4cm. In any case, it looks like the page just offers you the chance to pick from a variety of designs, but they do explicitly offer custom engraving from your own image; under "add your personalization" you should give the size head you want and then say, "I have an image for a design, please message and I will send it to you."

They reached out about 24 hours after I purchased, through Etsy, and asked for the image; this is what I sent them, also through Etsy's messenger function, and the result:

I'm given to understand that they generally have to retrace the image regardless of size or quality to turn it into a vector, so they can take a fairly low-res image and do a lot with it, but I tried to give them as clean a JPG as I could to work with. While the result is not absolutely identical, for an engraving smaller than a quarter it's pretty damn accurate and I think most of the minor shifts in design were for clarity's sake.

But I was pleased it really was that simple -- I bought the item, they said "Please send the design", I sent it, they confirmed they had it, and I didn't hear from them again until they sent me the shipping notification. Potentially if you don't have as clean an image they might need to talk to you about issues with the design, so I would make sure you have a very clean-line black-and-white image to send, but they seem pretty competent and they do good work.