I can only give you my experience with a 9-week old pup on a one-way from Chicago to NY to pickup our V from the breeder.
I believe most airlines allow for pet travel as long as it fits under the seat and is in "an airline approved carrier", which I purchased in Amazon for $20. There's obviously dimension requirements, I simply forget what they are for under the seat airline approved carriers.
We flew Southwest (which is more pet friendly than some airlines in fees, etc) and contacted them via phone support to inform them that we will be traveling with a pet/puppy. It was simply to have them note my ticket, probably overkill but did it anyways. At the time of arrival at the airport we purchased a boarding pass of our V which was $90 I believe and off we went. Pretty straightforward, but DEFINITELY contact United and discuss your plans because I am sure every airline is slightly different.
To that end, I think your only issue you may face is will a 14-15 week old V and 16+ weeks once you return from vacation to fit in an airline approved carrier. That may be tricky. My 9-month old fit and seemed to fit comfortably. Luckily TX to CA is not a terribly long flight so you have that in your favor.
Some small pieces of advice that was given to me, and many are obvious but bears repeating. Do not feed the pup the morning of the flight. Show up to the airport early, we hit a little snafu when we arrived to Southwest and took about 20-minutes to sort out due to the puppy (long story not worth sharing). We walked our V for a short time before actually entering the airport so he could eliminate and get tired. We put a nice adsorbent towel on the bottom of the carrier for obvious reasons. And the one thing I was told, and it held true, is once you're in your seat and the pup feels and hears the vibration of the plane they due tend to go to sleep especially once you take off. Our V literally slept the entire time and did not hear a peep from him from the time we boarded the plane to the time we got off....maybe we were just lucky but our airline experience with was very positive for us and our V. And I am definitely glad we made the decision to fly out to IL to personally pick him up and fly back. Well worth the trouble and small overall expense so he would not have to deal with other means of airline transportation under the plane and picking him up at the airport....fwiw.
I hope this help and please let me know if I can answer any other questions. And like I said, your biggest challenge right now is his size and ability to fit in the carrier. All other concerns can be sorted out IMO.
All the best with your new V.