I am intending to take pictures of ticks which should look like the one in this picture or even more detailed if possible. Taking into account that the ticks would be motionless, their size would range from this small to this big, and the fact that pictures would be taken using a tripod in a room with enough light, what type of camera body and lenses should I look for?
What you are looking for is macro photography and those ticks are actually so small that you must get something with a really high magnification. I know that some people use microscopes with a camera attached but I have no idea how that works, so I will answer purely from a standard camera and lens perspective.
The highest magnification lens currently available is the Canon MP-E 65mm F/2.8 Macro which has a 5X magnification. You can combine it with any Canon DSLR, even the most basic one such as Rebel SL2, all the way to an 5DS R which will give you much higher resolution, so it will capture more details even though magnification is the same.
Your biggest problem with illumination is avoiding the shadow of the lens to appear on your subject, so you should buy some small lamps that you can place very close. Canon actually makes a macro lens with built-in illumination but its magnification is 1.2X which will not be enough for those small ticks, it is the EF-M 28mm F/3.5 Macro which is compatible only with Canon mirrorless cameras.
There exists a 5X magnification lens with illumination by Yasuhara but I have never seen it as it is not widely available. This one is compatible with Micro Four-Thirds cameras but is much more manual as even the aperture must be set manually. With the Canon MP-E 65mm, you must still focus manually but the camera can take care of exposure.