Impeachment is a power that resides with the house, but there are procedures and constitutional laws to be followed in order to impeach (see the OP).
** bzzt ** wrong
The Constitution provides NO GUIDELINES for an impeachment inquiry, none whatsoever. It only says that the House can impeach the President and selected officials, then the Senate holds a trial which is presided over by the Chief Justice of the SCOTUS, and that's it. There is no requirement for any formal vote on an inquiry, and no expansion of powers once one is held.
Impeachment is not a criminal proceeding. The POTUS has no due process rights that specifically apply to an impeachment. There is no Constitutional or statutory requirement for any sort of full house vote on proceeding with an impeachment. The POTUS has no right to "face his accuser" during an impeachment. Congress already has sufficient oversight powers to demand documents without needing any warrants. The House does not get any special powers after such a vote. Precedent, which barely exists, is not binding.
Currently the House is not allowing for anyone other than Democrats to subpeona and investigate because of their unprecedented behavior (the Dems are making a power play).
News flash! That won't change based on any sort of formal vote. Right now, the investigation is run by a bunch of committees, which will submit their findings to the Judicial Committee, which will decide whether to advance a vote on impeachment to the full House.
House leadership (Pelosi and Schiff) are, I believe, intentionally making this a political issue for political, 2020 election purposes.
What are you, new?

Trump is also trying to use this for political purposes. I'm all but convinced that this letter is not even trying to be legally sound, it's a political move that would not work at all in a real court.
Trump has turned the tide of this entire narrative and called out the Democrats - bring a vote on the floor of the house (Trump and the R's are making a counter move).
lol
Trump hasn't "turned" anything. He's throwing daily Twitter Temper Tantrums since day one, and this is just an extension of that tactic. Best of all, his refusal to cooperate -- issued less than a week after Trump
personally vowed to cooperate with the inquiries -- is obstruction, which risks becoming yet another impeachable offense.
Last but not least: Trump's legal teams basically much suck ass. They rush their cases, don't do proper research, make awful legal claims, and are frequently undermined by Trump's public statements. As a result,
the Trump administration loses over 90% of its cases in the courts; most administrations only lose 30%. And it's not partisan, either -- many of the judges who shoot down administration policies are appointed by Republicans.
Roundup: Trump-Era Agency Policy in the Courts – Institute for Policy Integrity