Learn French: Les marionnettes que j'ai vues chanter sont drôles.

This is the core of the current lesson! With verbs of perception like voir followed by an infinitive (chanter), the past participle agrees with the preceding direct object ONLY if that object is performing the action of the infinitive. Here, the preceding direct object is que (standing for the feminine plural les marionnettes). Since the puppets are the ones doing the singing, vu takes the feminine plural ending -es to become vues.

What if the sentence was 'The songs that I heard sung' instead?

If the preceding direct object is not doing the action of the infinitive, there is no agreement. In the sentence 'The songs that I heard sung' (Les chansons que j'ai entendu chanter), the songs are the things being sung, not the ones doing the singing. Therefore, entendu does not agree with chansons.

Could I leave vu without agreement here, since the rules for perception verbs are so tricky?

In standard formal grammar, no: you must add the -es because the puppets are performing the action. However, a 1990 spelling reform made it officially acceptable to leave the past participle uninflected before an infinitive, so que j'ai vu chanter is generally accepted in modern everyday usage. Still, you should know and practice the traditional rule for B2 level exams and formal writing.

Why use que instead of qui since the puppets are doing an action (singing)?