No, you literally cannot. Synthesizing a stretch of DNA is trivial, and what you can send away for [1].

Building a functional virus is not even remotely a turn-key operation. Assuming you could get the genome, expressing that genome in a way that results in infectious virus is non-trivial.

[1] Even then, the companies will rarely give you long enough reads to make a full genome scaffold. It looks like there are two companies that will do 50-100kb assemblies as a service today, but I guarantee you they're going to look at what you're trying to make, and who is asking them to do it.

Would they not be like “hmmm this one looks kinda similar to a known virus”

They really just synthesize anything for you?

Yes in much the same way you can literally build a rocket and go to the moon.

You may be skipping a few of the involved steps.

One of the easiest and most effective AI regulations would be to simply require all these labs to run all incoming requests through a suite of the most advanced models asking if the request is likely to be a pathogenic agent.