I know that shields are usually soldered to headers so they can be attached to the board with the Arduino. I have a tremor, which makes soldering impossible. Are there ways of connecting PCBs without soldering?
Sparkfun has a lot of sensor boards that use their Qwiic Connect System. Those you can just plug in, and even chain together, with their cables with JST connectors.
Yes, there are ways. There is a system called "Press Fit" where the pins have a bulge in one end of them. This grips the inside of the holes in place of soldering.
Of course, these depend on the holes being just the right size, and finding the right connectors could be tricky... so... your mileage may vary.
But nothing is better than getting someone to solder the pins for you. You could teach one of your younger relatives to solder, for example...
Boards with headers can be connected using jumper wires or hookup wire. If there are no headers then I don't think you will be able to get a reliable connection without soldering.
The RedBoard you linked to uses headers, so you should be able to use jumpers with it, but I suspect you are out of luck with the WiFi shield. You'll probably either need to find a version that is built using headers or ask/hire somebody to solder headers onto it for you.
As for the center portion of the shield you should probably just skip that part and use a breadboard, which uses jumpers or hook-up wires.