Connecting new subwoofer to old stereo

First off, what model sub do you have? It'll make this easier as we can see what input options it has. If it is a passive sub (no onboard amplifier, sometimes these are included with home theater in a box systems), just connect the speaker B leads to the subwoofer and engage speakers A+B on the receiver. This isn't ideal as the sub will be receiving full range audio, but it'll work.

It gets a bit trickier if you have a powered sub (it has its own amplifier). If the sub has speaker level inputs (the type that accept speaker cable, not the RCA plug type), then you could use the B Speaker outputs from your receiver for the sub just like above, but you can use the crossover (if available on the unit) in the sub to filter out the higher frequencies. Just engage both speakers A+B on the receiver again. If your sub doesn't have that connection, meaning it only has RCA style plugs, there is not a real practical way to use the sub. The tape deck outputs are sent at a constant level, meaning that you wouldn't be able to control the volume of the sub from the receiver as the receiver is always putting out full volume. You could adjust the volume on the subwoofer's amp, but it won't track the volume of the main speakers. Meaning if you turn the volume up or down on the receiver, you will need to manually adjust the volume on the subwoofers amp as well. Technically this could work, but in reality would probably be so cumbersome it wouldn't be feasible on a day to day basis.

Running dual subs in a main system isn't a bad idea in most cases. Proper placement can help with room modes/nodes, and it is an easier process, just buy a y-splitter and run the lines.

Running the sub on your older AVR, you could try a speaker level to line level converter, like this one, try building a converter yourself, or buy a new AVR.

The device linked to was just an example piece, here are more, and another. The concept is that you would use one of these to take speaker cable, from your "B speakers" output on your receiver, plug the speaker cable into the converter, and then the RCA connections from converter to your wireless transmitter. Then select on your receiver both speakers A+B. Speaker A are your speakers, and in this case speaker B will be the sub. This is the only (easy) way to be able to control the volume of the speakers and sub together with your particular models of receiver and sub, since the receiver doesn't have any preouts/LFE out and your sub doesn't have speaker level inputs. You would only need to adjust the volume on the sub amp during set up, and then whenever you change the volume on the receiver both the speakers and sub will change their volume together. (We'll call this option A) This does require the B speaker outputs on your receiver, if you are running two pairs of speakers from this receiver as it is, then this will not be possible. If that is the case, you'd need to either buy another receiver or use the sub in your main system (sorry if that comes across as rude, its not meant to, just being practical).

In regards to the tape out question, as I mentioned above, technically it can work. The problem is that the tape deck output is not volume controlled. So if you decide to use that option, you will not be able to control the volume of the sub through the receiver (the receiver will always be sending full volume signals out of the tape deck connections). So if you want to change the volume, you would need to change the volume on the receiver for the speakers, then walk to the sub and change the volume for the sub through its own amp (we'll call this option B).

Technically this (option B) would work, but to me it seems it would be far more of a day-to-day hassle then buying an inexpensive line converter and using that (as in option A).

You're very welcome, glad to help. The primary differences are the form factor/appearance, whether the converter has pre-attached leads,and whether the converter has resistors in it to protect equipment from wattage received over the speaker level inputs. I'd look for something with a wattage claim to be safe, but I personally wouldn't worry about needing a very high figure. Since you won't need to drive the converter to max volume/level the amount of power being sent to the device isn't going to be that much. That said, I'd just find one that lists 15-20 watts per channel, has speaker level leads already attached (to keep things easy, you can just plug it directly into the receivers speaker posts), and is a form factor that works for you.

In case you want to do any further reading (not really necessary as you've got the concept, but in case you're bored)
AVS Thread 1
AVS Thread 2
More technical AVS thread
Schematic in case you want to build your own.