Learn Polish: Weźcie ten ciężki bagaż.

The ending -cie in weźcie tells us that the speaker is talking to more than one person (the plural "you"). If you wanted to tell just one person to take the luggage, you would drop the -cie and just say Weź ten ciężki bagaż.

What is the dictionary form of weźcie? It looks very irregular.

The dictionary form (infinitive) is wziąć (to take). It is a highly irregular verb in Polish, which is why the command forms weź (singular) and weźcie (plural) look so different from the base word. This is one you just have to memorize!

The luggage is the object being taken. Why don't ten, ciężki, and bagaż change their endings for the accusative case?

They actually are in the accusative case! In Polish, for non-living masculine nouns (masculine inanimate) like bagaż, the accusative case looks exactly the same as the dictionary form. So ten ciężki bagaż is the direct object here, but its spelling doesn't change.

Are the words for "this heavy luggage" plural because we are talking to multiple people?

No. Even though the verb weźcie (you all take) is plural, the object ten ciężki bagaż (this heavy luggage) is singular. The noun's form depends on the luggage itself, not how many people are carrying it.