I have a old kitchen kitchen faucet with attached sprayer. I want to install a Culligan undersink filter which comes with a water faucet that REPLACES the sprayer. I have very limited room under the sink to work in. I cannot even get any normal open end wrenches/adjustibles to work with limited space.

How can I make the sprayer not usable? I assume there is always water pressure going up through the sprayer hose, so just cutting it off is not an option.

Any help would be appreciated!

If the space's tight then purchase a basin wrench,(see image) for those hard to reaach nuts and bolts . While you're under the sink look for hose connection. Unscrew hose and sprayer and replace with a 1/8th brass plug. Don't forget to TeflonTape the threads. You may now take your basin wrench and remove the nut that secures the sprayer head and install your new water faucet.
Good luck, Tom