I suspect Tyson is right about the old holes being from termite treatment (although they should be 1' apart for that, so it would have been a crappy, ineffective treatment job). If they were from previous jacking and the slabs now need jacking, that would kind of indicate that jacking isn't a good long-term solution (and there are lots of reports that such is the case).
Aside from longevity issues, jacking isn't as easy as it looks in the online videos. Getting good results takes a lot of practice (you need to anticipate where things will end up when the material stops spreading or expanding and stop pumping ahead of time, you need a feel for how slabs will move, even just keeping thing neat takes practice). If you're talking about foam jacking, there are lots of ways that can go wrong, and the stuff is a mess to clean up.
I don't know if a license is required for using the bulk chemicals, but the reason you may not find rental equipment is the likelihood that amateurs would return the equipment in a condition that it would need to just be disposed of.
I would recommend solving the problem a different way. Leave the slabs where they are and add a layer on top. There are high strength topping materials. Make the top layer slightly wedge-shaped to achieve the slope you need.
- If the slabs are level and you're just getting runoff in both directions, it might be enough to just add a small cove at the house that keeps water from running off on the house side.
That looks like a narrow sidewalk and then not much distance to the property line. From the picture, it looks like maybe 4 - 5' from the house to the fence. Even if you get the water to run away from the house, a lot of that is still going to permeate the ground and get back to the foundation.
There are some things you could do to improve the situation. Pull everything up, line the area with plastic and put everything back. That would create a problem for your neighbor, though. Or pull everything up, excavate your foundation and seal it, then put everything back. That would be very expensive if you could even get the equipment into the space and had a way to deal with the mountain of dirt. Another thing those old holes could be is a previous attempt to inject "waterproofing" around the foundation (not effective).
The best solution to keeping the runoff from getting back to the foundation (after you deal with the slabs), is to bury a drain channel next to the slabs on the fence side. There are "kits" that combine pipe with surface drains. You trench next to the sidewalk and bury the system just below slab height. The captured water flows to an area far away from the house. If the ground has some slope, you can just discharge the water. If not, you can create a dry well. There are also some systems that hold the water and just release it slowly into the soil when the soil is dry. Those need to be sized for the volume of water, the extremes of rainfall, type of soil, etc.