You need to find someone with a well-tuned bandsaw. I really doubt you can get 5 pieces. Getting 4 pieces will require a fair amount of luck.
That luck will depend on the board - how perfectly straight and flat the board is. Both surfaces must be flat and parallel over the entire 4.5 foot length with no cupping, twisting, bowing, surface cracks or checks, etc. The wood type matters, too. If the wood is very hard with gnarly, twisty grain, you may lose a little bit as the blade tries to follow the grain.
That luck will also depend on the bandsaw. It must be perfectly turned with zero drift and no blade bowing during the cuts. The blade must be perfectly parallel to the fence so the top and bottom of the cut are exactly the same thickness. Blade type, thickness, and sharpness matter.
How much sanding (or planing) will be required to remove the saw marks? The middle boards will require blade marks removed from both sides.
If the board is perfect, then you are starting with 7/8 = 14/16ths inch. Getting four 1/8 inch boards subtracts 8/16ths and leaves 6/16ths. For myself, I figure a generous 1/16th inch kerf for my bandsaw. Three cuts at 1/16th per cut subtracts 3/16ths and leaves 3/16ths. Assuming the outside surfaces are flat and perfect, that leaves you with 3/16ths to sand the blade marks from six inside surfaces (1/32 each surface).
Getting four 1/8 inch boards from a 7/8 board would be pretty good.