What type of switch machine? If a Tortoise, you're going to need some music wire size .032 or so to replace the wire that comes with the machine. My HO scale home layout is built on hollow core doors covered with 1 inch of sound board, foam roadbed, and track.
It would help if you told us what brand and scale of turnouts you're wanting to power before we can explain in more detail on how to proceed.
Okay, the power supply doesn't matter unless you're going to be using Tortoise switch motors and a stationary decoder so you can throw them with a DCC controller.
First, drill a 1/4 inch diameter hole directly under the center of your turnout's throw bar. I then use a tapered drill bit to enlarge the hole from the bottom so the music wire has enough lateral movement and doesn't jam on the sides of the hole.
Using the throw wire template provided with the Tortoise, I bend a new throw rod. The hole in the Tortoise into which the end of this wire is inserted needs to be slightly enlarged.
I fasten the throw rod to the Tortoise and work the rod up through the hole in the turnout throwbar, fasten the motor temporarily with double-sided white foam tape, and cut the wire off from the top, leaving about 1/2 inch sticking up while I adjust the alignment of the motor. I power the switch motor to one side or the other and adjust its position according to the instructions with the machine. When I have it in the right position, I drill four small holes and fasten it permanently in place with small wood screws. Then I clip off the remainder of the wire sticking up from the throwbar.
Practically all of this is covered in the Tortoise machine's instruction sheet.
The above assumes a tortoise switch machine will be used - which ain't necessarily so. I use a variety of devices to move points, Tortoi not among them.
The linkage I use is a modified form of Anderson link, based on having a narrow tube (1/8 inch or less in diameter) drilled vertically through everything from tie-top level to 1/4 inch below your lowest subgrade plywood. The tube is 1/2 inch (not super-critical) from a hole in your switch throw rod - which can be molded in or drilled for the purpose. The hole might be between the rails, but outside the rails is safer. It can be either ahead of the points or between the throw rod and point pivots. The line from hole to tube should parallel the nearest rail when the points are centered.
(Special note for Atlas switches - the horizontal hole in the throw rod may be used, but is awkward to get the throw wire into unless the turnout is loose from the roadbed and adjacent trackage. You have to 'needle the thread.')
The next item is a piece of springy steel wire, bent into an L shape, long end down the tube, short end (with a second bend to engage a vertical hole) to the throwbar. Below the subgrade, that wire is bent in whatever direction is convenient for connection with whatever is going to throw the switch. I bend a loop in mine, then attach a length of heavy monofilament fishing line leaving free ends in both directions. The line and the wire to the tube should be at right angles when the points are centered.
One end of the fishing line goes to a weight (a couple of stripped machine nuts, fishing weight or whatever) which holds the points in the 'normal' position. The other end may connect to a manual throw device (which can be an electrical switch, a Caboose throw mechanism or even a barrel bolt) or an electric switch machine - located at the table edge where it can be wired without having to get under the benchwork. Line guides for the monofilament are small screweyes screwed into the plywood sub-roadbed from below. I personally use toggle switches for manual throws, and twin-coil machines for switches which must be operated from remote locations.
When mounting a twin-coil switch machine, the only requirement is for the armature to move in a horizontal plane. The machine itself can be mounted vertically (on edge) to a mounting board just behind the (removable for access) fascia.
Chuck (modeling Central Japan in September, 1964 - Analog DC, MZL system)