Can I use an inner tube that's too narrow for the tyre?

Yes, you can, as long as you don't go more than one size smaller, or the differential is not too extreme. A 1.25 in a 1.5 is fine. A 1.25 in a 2.25 is likely to have issues with flats.

Your LBS will have tubes. Are they that bad that you won't even buy tubes from them?

You may be perfectly OK, or may experience several possible problems:

• The area near the valve on the tube is the stiffest part, and does not expand as easily as the rest when over-inflated. This may result in a slight depression in the tire near the valve that you will feel on every revolution (especially if you start thinking about it). Not a significant reliability issue, but it can be a comfort issue.
• As the tube expands in its width from over-inflation, it expands even more in its overall circumference. The tube can, before it gets "locked in" to the sidewalls of the tire, expand enough that a portion of the tube telescopes on itself, in the short term creating a lump, and in the only slightly longer term creating a stress point that will result in a fairly sudden deflation of the tire.
• And obviously, the tube is being placed under more stress than it's designed for, and it's apt to simply fail, especially near the valve.

I wouldn't hesitate to use the (slightly) wrong size tube for a relatively brief period, as an emergency measure, but I wouldn't regard such a repair as trustworthy over the long haul.

Adding to the thoughts above: I had a puncture yesterday on my ride home and used way smaller tube (had the wrong size with me and got another from a fellow bicycle commuter; both tubes were for racing tyre size where mine is 700x33c). First time I tried the tube immediately got damaged and deflated completely - even before I had the wheel back in the frame. The second time it lasted about 100m. So, probably fine to go just one size down but not too much.

ANSWER: yes smaller tubes can fill larger tyres, but they may fail quickly and unexpectedly.

I had this exact same situation. I am converting a road bike to urban, and the largest size tire I could fit was a 700x28, the road bike has 700x23.

When I got the old tires off and pulled the tube it was sized 700x23-25. I had changed the tires and reused the old tubes and thought a 10 mile(16 km) ride ought to be safe. 5.2 miles (8.3km) later I was calling my daughter for a pickup. The front tire just went flat. No bump, rock, pothole, nail...just went flat. The rear tire stayed up the entire time, but I am changing its tube too and will keep that one in reserve. I am thinking it was the valve stem.

I've had a 19 to 23 tube in a 32mm (700) tyre for a while now and no issues. The 32 is a Michelin Air Max tyre with a very thick sidewall, so that is probably helping.