A good question, well-deserving of a meaningful and thoroughly considered answer.
I think they do, at least some of them, and I have three reasons.
1. The only reason for an LMD to exist is to represent a person who can’t be there (because of death, other commitments, laziness, etc.) but LMDs are most often used in subterfuge. It seems like it would be a major design flaw to give them no genitalia, since that’s a super easy way to find out if someone is an LMD. I mean even if they didn’t originally, someone would have worked out that LMDs Are Smooth Like Barbie Down There and then before every secret meeting or whatnot people would just do Pants Checks to make sure the person they were talking to was real. So you’d have to START putting genitals on LMDs just to circumvent the Pants Check. But I think they probably did from the start because:
2. It can’t be that expensive to just plop a dick on one, given that they’ve already spent all that money on programming and lifelike skin and such.
3. Also, we know that Doombots have genitalia because Clint once fucked one while it was pretending to be Wanda. It’s not that you need genitals to have a good time, and they come in a wide variety of configurations, but I’m pretty sure even Clint would have had his suspicions aroused by Wanda having no genitalia at all. My point in bringing up Doombots is that if Victor Von Doom put genitals on his robots you can be sure SHIELD and Stark Industries would as well, just to make sure they’re keeping pace with market trends.
3a. I am also pretty sure that we have at least a couple of examples of LMDs being intimate with people in various “your entire reality is a lie and everyone you love is a robot” scenarios over the years, but I can’t call any to mind, so I’m going with the “Keeping Up With The Doomses” theory.
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