Many companies offer that. In the local area they have e. g. mail-like stations where you give out your order and they give you what you need (e. g. if it is a plasmid, it is mailed into storage boxes often on the same day or next day).

"Not keeping it on their side" ... well. If they sequenced it, they have data in their computer, right? How could you avoid that? I don't see this as possible, it depends on trust (and whether you really care about that).

> What service would I use / you can recommend?

I won't recommend anything as I do not want to be an ad-amplifier, but my personal rule of thumb is that those companies that are affiliated with science clusters, are often chosen because they offer high quality (and to some extent because of corruption e. g. xyz knows abc, but this is IMO the smaller part, usually it is quality/speed/ease of use).