"The invoice for June" is only correct if the sale of the goods or services that are invoiced happened in June. If the invoiced was issued in June, but the sale of the goods or services occurred outside of June, then it would be "the invoice from June". "The invoice of June" is only correct if it is referring to an invoice where a person named June is the vendor or purchaser; but "June's invoice" would definitely be more common in this scenario and it's a fairly rare scenario because June is a fairly uncommon name.

The invoice for join. We use for to talk about a purpose or a reason for something. The payment for June is the reason for the invoice.

"Of" isn't technically incorrect, but "for" is much more common and sounds much more natural.

The invoice for June.