Would Americans say:
He sat down 9 feet from me.
He sat down 3 yards from me.
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Feet is the more common, conversational usage in the U.S. Your speaker would say he sat nine feet away from me
Yards are often used to describe particular things that are traditionally measured in yards. Sports often use yards to describe distances. A football field is measured in yards. A golf hole is measured in yards. Foot races used to be measured in yards (meters now).
My initial thought was that feet would be much more commonly used. Ngrams backs that up.
Either is idiomatic.
We tend to like more readily visualizable distances, an inch before a foot, a foot before a yard, a yard before a mile. But we also tend to like smaller numbers and whole numbers. We would be more likely to say 5 feet than 60 inches, but 18 inches rather than one and a half feet.
Between three yards and nine feet, I suspect it would be some of one and some of the other.
I don't think an American would try to be precise as '9 feet' unless it was somehow important. In the context of sitting distance it seems unimportant to me so it's more likely they'd say 'roughly 9 feet away' or 'about 9 feet away' than anything else. Now even more likely is that they round up to 10 feet as then it becomes clear that it's an estimate. If the context is football then I can see yards being used instead but otherwise yards feels out of place.
Relatively small distances (such as 9 feet/3 yards) would normally be given in feet, as yards introduce too much "slop" (plus or minus amount) for comfort. 3 yards would technically mean somewhere between 2.5 and 3.5 yards (7.5 to 10.5 feet), which is too imprecise for most people when talking about such distances. 9 feet would be 8.5 to 9.5 feet, a precision that most people would be comfortable with. When you start getting up to longer distances (such as the length of a football field), the +/- precision loss due to using yards instead of feet becomes insignificant, compared to being able to use more convenient smaller numbers.
The American measurement system developed in a way that there is a bit of overlap between units. Your example is in one of those regions where either would be appropriate and not unwieldy. There is some dependency on the setting as to which unit of measurement would be more appropriate.
Yards are typically used for outdoor measurements where the distances are larger, but not large enough to get into the mile ranges. Yards are also used by people that tend to participate in outdoor activities where they will have to estimate distances in the region where using the yard as a unit of measurement makes sense. These activities will include football, golf, sprinting, and the shooting sports.
When the setting indoors the room sizes, unless you are a in a massive room, don't give much opportunity to give measurements in yards. If you were outdoors, I would favor yards. When indoors, unless you are in one of those aforementioned rooms, or wanted to add some of that outdoor flavor, feet would be more appropriate.
Using yards in place of feet when the situation is neutral will also make a person appear more outdoorsy or sporty, as the overlap in units allows for some personal preference, and the word choice would reveal a greater familiarity with that unit.
The reality behind this question is that the normal answer would be "He sat down 10 feet from me".
An answer like "He sat down 9 feet from me" or "He sat down 3 yards from me" implies an intent to diminish the actual distance and paint the picture that the individual sat down in a very close proximity to the person making that statement.
Objective measurement is a concept expressed particularly well in "the metric system".... er, SI. But most of the communication that we're doing (outside of the sciences and engineering) is going to be subjective. Let me give a handful of examples:
So, if I'm trying to express that an individual sat down 108 inches away from me I would tend to express it in a way that communicates what I'm trying to. Are we in the middle of a desert, nothing else for days? Then I'm going to express it by saying that they "were within spittin' distance". Are we in the movie theater? I'd say that they "sat down three seats down from me". Are we on a date? "She sat down just far enough away that we couldn't have a conversation, I won't call her back".
The same goes for larger measurements, as well. 100 yards? We know that because it's a football field, so we might use that as the unit. If I'm talking to people around town I'd say "Half a block away" or "three houses down", if I'm talking to people at the farm I'd probably tell them that "it's just a bit farther than I could chuck a baseball," unless I was talking to Mike, and then I'd say "You could hit it with a football," 'cause Mike likes to think that he's got an arm on him.
When giving directions, I don't care that you need to turn 1.347 miles after you get on the road, it's "two stoplights away". And the next gas station isn't 17 miles down the highway, it's "22 minutes from here", or "At the next Sonic Drive-In". And my sister doesn't live 610.7 miles from here, she lives (depending on context!) "9 hours West" or "The next state over".
It goes the other way, too. When I get a latte at the coffee shop, I ask them to add "just a drop of Hazelnut, I don't want it sweet at all". When I'm doing woodwork I complain that "I'm not done sanding, I can still catch it with my fingernail". And that's just a couple of the measurements that I've used since yesterday!