Unfortunately steam rockets would not be practical because they're not very efficient.
I don't know how much you know about rocket propulsion so I'll begin with a discussion of one of the most important metrics: exhaust velocity, the speed of the gas coming out of the engine's nozzle. The higher that speed, generally the more efficient the engine. There are other considerations, to be sure. For instance, if an engine that produces 500 Newtons (~110 lb) of thrust has a mass of 100 kg, that's not very mass-efficient!
The thrust produced by an engine is directly proportional to the exhaust velocity $V_e$ and the mass flow rate ($\dot{m}$), which is the mass of propellant fed into (and coming out of) the engine per second: $$F=\dot{m}V_e$$
You can see that as $V_e$ increases, the more thrust you get from a given rate of propellant use. Or, for a given thrust you need, such as the thrust required to lift you from the surface, the higher the $V_e$ the smaller the propellant use rate required. And that's important.
(Aside: instead of exhaust velocity rocket engineers usually refer to specific impulse, which is the exhaust velocity divided by the gravitational acceleration at Earth's surface)
By the term "steam" rocket engines I mean rockets expelling steam from their nozzles after heating it by means such as solar heating or electrical heating or some such, not by direct chemical combustion. I make this distinction because what came out of the Space Shuttle's main engines was steam, but it was created by direct chemical combustion in the engines' combustion chambers. Those engines were very efficient, with very high exhaust velocities. Steam rocket engines' exhaust velocities are much lower.
Why were Space Shuttle engines so efficient when steam rocket engines aren't? It's that exhaust velocity! And the exhaust velocity is a direct result of high temperatures in the chamber: for a given gas coming out of the nozzle (steam, in this case), the higher the chamber temperature, the faster you can make the exhaust go. On the moon, with solar concentrators or electric heaters, you might get the steam up to several hundred °C. Space Shuttle main engines ran at ~3,300 °C chamber temperatures! That's why they were so efficient.
Why would efficiency be a concern?
The more efficent, the less propellant you'll use up. This has several advantages, such as: your personal rocket-pack rig will be able to work with smaller tanks, making it less unwieldy; and your water mining operation and production plant can be smaller.
There are multiple engineering issues with steam engines. For one, when you heat the steam to several hundred °C, unless you carry a heat source with you, that steam cools due to heat lost through the tank walls, and it gets less efficient. If you have good insulation that cooling goes slowly; with not-so-good insulation it can go quickly. Heat sources can be heavy and bulky. If you're going to have some fixed infrastructure to heat the steam (call it a charging station?), then once you take off and are separated from that infrastructure, you have a finite amount of time to finish your task and get back to it. If you stay out too long, even sitting on the surface somewhere, the steam could get too cool to produce the needed thrust, and then you're stuck! Another is that the steam engine is an example of what is called a "cold gas thruster", an engine that uses a tank of pressurized gas without any kind of internal combustion to heat it. There are other gases, such as nitrogen or helium, that are much easier to use in a cold gas thruster: they don't have to be really hot because their liquefaction temperatures are really low and condensation in a nozzle won't be a significant problem, they aren't as reactive as water, and so on. (But there isn't much of those on the moon. It might be better to use O$_2$ extracted from lunar rocks, despite its reactivity.)
Efficiency is desirable also because there is a finite amount of water on the moon. Estimates run from half a billion to a couple billion metric tons, which sounds like a lot. But if you use it in ways that disperse it, your water production infrastructure must be sized to replace it, and eventually that resource will be gone. Rocket engines whose exhaust is steam disperse water, the great majority of which is lost to space. If you're going to use water in a way that disperses it, it's best to use it in the most efficient way. At least, until you can mine the asteroid belt for a source of water more efficient than lifting it out of Earth's gravity well.
