Is it "under 10 people" or "fewer than 10 people"?

"Fewer than 10 people" is the precise form. Under reads as a spatial word, not a count.

Quick answer Canonicalizes to Under vs. Fewer than

Can I say "under 50 words"?

i · AnswerOne line, no lecture

You can, and most readers won't blink. Careful prose prefers "fewer than 50 words."

ii · A little moreWhy this is the one to keep

Under is spatial at root, and editors who notice tend to replace it with fewer than for count contexts. For a word limit — where the count is the whole point — fewer than is tighter. Style guides differ; house style usually wins.

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Less vs. Fewer For things you measure versus things you count. Usage
Who vs. Whom Subject versus object — the pronoun doing it versus the pronoun it happens to. Usage
Lay vs. Lie You lay something down. You lie down yourself. Usage
That vs. Which Essential versus extra — a comma decides. Usage