This wiring MUST be in a receptacle box that is mounted in the wall. I just fixed a mess like this in another house. This is a wet environment, so there should never be exposed wires, even capped and taped.
If you were doing this correctly, the ROMEX would be inside the wall and pulled into the box. A receptacle would get the power but the other parts of the circuit would be connected inside the box too. The picture does not show where the ROMEX originates but it looks like it is from the top.
Here is what I would suggest:
1. Kill power to the feed ROMEX. If you don't know how to do that, STOP and get help.
2. Label power ROMEX versus other parts of the circuit.
3. Cut into the back wall so you can bury all the ROMEX wires.
4. Cut drywall for an "old work" receptacle box (plastic) no less than 10" above the floor of the cabinet.
5. Pull all the ROMEX wires into the wall and into the back of the receptacle box.
6. Connect power ROMEX to a GFCI receptacle (required by code), making sure you attach the WHITE (neutral) to the silver screw and the BLACK (hot) to the brass screw. Connect these wires to the end of the GFCI labelled "LINE." (The diagram should help with this concept but the receptacle is NOT a GFCI which you need on the left side. DO NOT put another GFCI on the right side )
7. (Can't tell from your pic) Now provide power and neutral to the other circuit(s) by attaching to the bottom screws on the receptacle (labelled the "LOAD" end of the GFCI).
8. Make sure your grounds are all attached together with one wire nut.
9. Install receptacle in the box and install the box in the wall. Attach a cover over the receptacle.
10. Remove bottom plate from disposal. Buy and install an appliance or disposal "whip" which will have a three-prong on one end. Make sure it is clamped to the disposal so it cannot be pulled out.
11. Plug in the disposal.
Lastly, if it were my house, I'd never use the damn disposal. They clog up too often. Never run pasta, potato peelings, coffee grinds, egg shells, rice down the disposal unless you want to fix a blockage.