So you have a 3-phase "delta" electric service coming from the utility to a well. This is non-residential, clearly. You want to tap the supply there.
This is going to be complicated.
Your tap will need overcurrent protection separate from the well pump, meaning there'll need to be a (sub)panel there if there isn't. This makes the work too complex. This isn't "tap a 120V receptacle or lamp". It'll blow right past the "trivial" exception for non-qualified persons, and obviously the "homeowner-occupant of a single-family home" exception doesn't work either. A pro electrician needs to do the work.
For all we know, this pump is 480V delta... in fact I'm not sure why it wouldn't be.
So we need a ton of information even to advise, and it's almost certainly going to cost some money. And this is my problem. If I'm having you do a power cut to insert a service panel to support more than just the well, we are blowing right past the cost of a battery/solar system, given that we're talking about under $200 of batteries assuming you just grab random deep cycle wet cells, cheaper is possible - and $50 of solar + charge controller. The gate vendor may have some pricy bundle, but that's not necessary.
I know you want a "hook my 2 wires here" solution, and I realize it can suck when there are 5 other guys all to happy to do the work illegally and unsafely (and that may not even work if the service is 480V).
Think hard about a battery solution.
Here is the secret to gates. Think about how much power your gate requires to move. It says 150 watts, yes? Well, if you ran the gate motor for an hour, that would be 150 watt-hours. Except the gate motor only runs for a few seconds in reality, yes? So less than 1 watt-hour per opening - this is a tiny amount of power actually.
A typical car battery is 1000 watt-hours (or 1000 openings), just for comparison.
As such, this application lends itself very, very well to batteries and solar, and then you can forget about the shady AC power connection.
Once when I said this, a fella got real mad and said "the HOA won't let me put up solar panels". This guy was imagining a huge solar panel array big enough to shade a car...and would not let go of that picture. I was thinking more like one the size of an iPad, stuck vertically on the side of the motor lol. The HOA would never be aware a solar panel was there. Which would be 10 watts, which would refill the battery with enough power for 50 openings on a bad day.
Anyway, the gate manufacturer is way ahead of me. They carefully chose "24 volts DC" as their operating voltage. That's a mighty odd pick, isn't it? Why not 120V? The reason is they are expecting you to put it on battery. It's just such a naturally useful application for it.
You DIY that by wiring two 12V batteries in series, or just choosing a 24V battery which might be lithium. To charge, you use a solar panel of almost any size ($20-ish) and a solar charge controller designed to charge the type of battery you use. I'm fond of Morningstar; a top brand. Sort of the Snap-On of solar equipment.
If the customer doesn't like it, I would just slap in a battery/solar "temporarily" while you run down the AC power question, and then let them have awhile to realize battery really does work.