House Rule XI(m), p. 19, states the power of committees and subcommittees to issue subpoenas.
(1)…a committee or sub-committee is authorized (subject to
…to require, by subpoena or otherwise, the attendance and testimony of
(3)(A)(i) Except as provided in sub-division (A)(ii), a subpoena may
be authorized and issued by a committee or subcommittee under
subparagraph (1)(B) in the conduct of an investigation or series of
investigations or activities only when authorized by the committee or
subcommittee, a majority being present.
A subcommittee could therefore authorize the chair of the subcommittee to issue subpoenas with no vote at all. Otherwise, the (sub)committee must "authorize" a subpoena, which conventionally means "vote on the motion". There is no requirement that the entire house must vote on subpoenas. There is no question that POTUS did not obey the demand of the House subcommittee.
The argument that will most likely be proffered is not that the House violated its rules, but that the House rules exceed constitutional authority in issuing the subpoena. As pointed out in this analysis, this is not a well-established and settled question of constitutional law.
There is no rule that requires a full House vote on all subcommittee rules established by a subcommittee that forwards articles of impeachment to the full House. That is, a subcommittee does not have to get prior approval of the full House in order for a subcommittee to conduct business and recommend an action to the full house. Instead, the House simply votes yes or no on the particular articles, following the rules for House votes.