There's absolutely no sex in this play. Unless you count the fact that Henry accuses Bushy and Green of having had sex with King Richard. Check it out:
You have in manner with your sinful hours
Made a divorce betwixt his queen and him,
Broke the possession of a royal bed
And stain'd the beauty of a fair queen's cheeks
With tears drawn from her eyes by your foul wrongs. (3.1.1)
Basically, Henry Bolingbroke implies that Bushy and Green seduced Richard, politically and sexually. If you ask us, the accusation sort of comes out of nowhere. We all know that Bushy and Green gave Richard a lot of really bad advice when they were his counselors, but does that mean they also seduced him sexually? We can't find any evidence that they did, so it seems like this little sex scandal is just a convenient way for Henry to get rid of Richard's allies. Besides, did you see how devastated Richard and his wife are when they're forced to part as Richard is led away to prison? Richard certainly doesn't act like a guy who's let someone else come "betwixt his queen and him."
Brain Snack: Shakespeare probably got this whole "Richard is a cheater" idea from two different literary sources for his play. The first source is Holinshed's Chronicles, where Holinshed writes that Bushy and Green went out and arranged for Richard to hook up with a bunch of women. It's also possible that Shakespeare borrowed the idea from another play that was popular at the time. In Christopher Marlowe's Edward II (c. 1593), King Edward seems to have a sexual relationship with a guy named Gaveston, who most definitely comes between Richard and his queen (Edward II, Act 1, scene 4).