Is there a way (other than collecting water and heating it on the stove ) to get hot/warm water from the tap without having to alter existing plumbing or electrical wiring ?
Check out the 1 or 2 gallon point of use heaters that install under the sink. They run on 120VAC and are fairly easy to install with minimal plumbing changes. There are also above sink mounted hot taps that mount into the sprayer hole in the sink. These are usually used for hot water for coffee/tea etc. Both of these type products are available at your local home improvement centers.
Here are several examples: http://www.lowes.com/Plumbing/Water-Heaters/Point-of-Use-Water-Heaters/_/N-1z11qho/pl#!
Small under the counter HW Heaters usually come with a stranded 3 prong plug. Drill a hole to the basement under the sink trough the cabinet and plug it in with an extension cord. Not exactly my preferred method of wiring something, but I think even in Chicago (anal codes) it would be permissible so long as it is plugged into a GFCI. Even if it was hard-wired it still should be on a GFCI breaker as it will be in close proximity of water. (code is 6ft from water requires a GFCI)...FYI all non-dedicated circuits in your kitchen should be GFCI protected. (the ones that are assessable, not the one for your fridge\stove cause GFCI's break a lot)
You need an extension cord and two flexible supply lines. If there is only one pipe coming in (cold) then you will need a new valve with 2 take offs or a 'T' and some nipples if you know how to pipe fit.
From the new valve or 'T' one goes to the cold on the faucet, the other to the 'in' on the new tank. The 'out' from the tank goes to the hot on the faucet.
Assuming drilling a hole gets to somewhere with an outlet this could be done in about an hour providing all the correct sized supply lines are on hand, a T, 3 nipples, a good heavy cord, and of course a HW tank. Put a pan under it for when it starts to leak in a few years.
-If that sink has power under it for a disposal everything you need is right there for you. Assuming the switch for it is a button on the sink and not on the wall. Meaning it needs to be on a non-switched circuit or it would only work when the disposal is running. If you have EMT (conduit) pulling some wires from the switch to a new outlet below the sink shouldn't be that big a deal. If its romex your kinda out of luck unless its 3 wire or more and one of the conductors isn't being used. Don't cheat and use the green or bare wire.\
-depending on which way the power comes from even if it is a switch on the wall it may be always-on power down there.
-stealing power from the dishwasher is an option, but don't be surprised when the breaker trips as both items have electric heat coils. I wouldn't even attempt this unless its a 20a breaker.