Your sentence reveals what traditional grammars call the subjunctive mood in English.
In the subjunctive, we use the unmarked infinitive of the verb, in this case (to) sit.
The chair was donated to the Museum on the condition that no one sit in it.
Thus, the word sit in your sentence is not the first person singular, present tense form of the verb (as in I sit there) but the unmarked infinitive or base form.
There are many verbs and expressions in English that reveal what we call the subjunctive. We use it to talk about conditions that aren't true, and in requests, wishes, and commands. (Note that on the condition that expresses a command, however gently!)
The subjunctive is fast disappearing from English (and many new grammars hold that it never existed in the first place.) It would be perfectly normal, correct, and expected to see your sentence written as:
The chair was donated to the Museum on the condition that no one sits in it.
There are many verbs and expressions that can reveal the subjunctive, and there are many lists of these verbs and expressions. One useful list, with a discussion and examples of the subjunctive, is here.