How does branching work in ROTC for National Guard soldiers?

Both are incorrect.
Most incorrect is the one telling you it’s the same as active duty. The Active Duty OML give the top 10 percent (distinguished military graduate) their first choice. After that, it is a complicated process where HRC tries to balance needs of the Army, Individual Prefrence, and ensuring quality officers are sent to every branch. (I had one Cadet get his 6th choice despite a 3.8 GPA at an Ivy, a 298 PT test, and a recommendation from his cadre and PMS to branch his #1, Infantry).
National Guard is completely different, and has absolutely nothing to do with the National OML or HRC (or even NGB for that matter)
That said, just getting a letter of acceptance from a unit is also not actually how it works either.
Every state is different, you could have two units that do the same thing 5 miles away from one another, but over the state line, and they probably have never met, worked together, or know anything about the end strength their counterparts. Each state is completely independent from each other state. With this, every state I’ve seen uses an Officer Strength Manager, usually assigned to Recruitkng and Retention (a few assign them to G1 or JFHQ), and they manage the vacancies and projected vacancies of the State for Officers (and Warrant Officers).
Each state is structured differently. For example, if you really want to be an Infantry Officer. New Hampshire is primarily an Artillery state, and there is only one company. Vermont, less than an hour away, has a Company, the BN HQ, a CAV squadron (which could take an infantry officer as a PL), and the Mountain Warefare schoolhouse. If you only talked to NH, you’d have a seriously limited chance at Infantry. If you talked to the next state over? Much better chances that one of their slots is open.
Each state has a set and specific Manning document authorizing each of their units, and what Soldoers make up those units. If a state has a vacancy, and they like you, or no one has applied, or you are the first, or something, you can get accepted into the slot. If a state loves you more than anyone, you have the best OML score in history, and no one else is applying to the whole state...you still can’t get a slot they don’t have vacant.

EDIT: I am having trouble catching all my typos on my iPhone-I am a former Army 1SG and ROTC Instructor, later waived as a full AMPS. I really do know what I'm talking about, even if typing on my phone I screw up or autocorrect something dumb.

The SMSgt is completely right. The only thing I want to add to the conversation is that you can fill an O1-O3 slot as a 2LT. I'm in an O-3 slot currently as a 2LT.

Generally in the National Guard you have the choice of going into any Branch active in your state that you qualify AND has slots available in those units. I would look at the type of units in the state and then talk to the State Headquarters about which one had slots available. I never solicited a certain unit, the state assigned me to the one that had open slots. During my short stint as an Infantry Officer with the Ohio Army National Guard, they had a Separate Brigade with three full Battalions, so lots of 2LT slots. When I moved to Kentucky, the state had only one Infantry Battalion, but they had two Battalions of Engineers and a shortage of Engineer Officers, so I branch transferred. I think you will find most states will be heavy on the Combat Arms side, with Engineers coming in a close second.