You need a sawtooth to drive the X of the scope and to tune a VCO.
If you have scope with a sawtooth output, like the Tek tube scopes (535/545/547) you van use a voltage divider to bring down the amplitude (that is around 100V)
Them make some VCO and use the sawtooth from the scope itself.
Important is that you must keep the VCO output amplitude constant over the sweep.
The filter must be terminated at both in the right impedance ( that is not always 50 Ohm) because a different termination pulls the filter of frequency.
To detect you can use the SNA priniple and use a tuned receiver and only listen at the transmit frequency. In that case harmonics or f.i. RFI from BC, telephones, mains etc do not polute the measurement. That is a lot of work. Basicly that is a SA with TG.
Wobbulators use diode detectors. That is much more simple and handy in case of a filter in a radio. You can inject the signal somewhere before the filter or even mixer en detect it after it. In that case you use a high impedance setup ( a diode detector can be made close to 50 Ohm but that is not easy.
Important is a marker function or a very well defined sweep ( so you know f.i. That the screen shows 1 MHz /div. It is very very easy to make a perfect filtershape at the wrong frequency. So i alway combine adjustments in radio filters with a TRMS meter at the audio output to check if the filter is at the right frequency and check if the 3dB points are too. Then the wobbulator can make the passband smooth and the skirts as steep as possible.