There is another phrase: "Sit up". A teacher might use this when asking one or more students to sit "properly", in other words to sit upright in their chairs rather than slouching over their desks.

You should not say "sit well" to mean "sit down properly" because "sit well" is a common English expression that means to be uncomfortable about a situation. Example: My daughter is dating a man 15 years older than she. That doesn't sit well with me.

If you're talking to children in a classroom you can remind them to 'sit nicely'. Sit well doesn't sound right.