Connecting anything to the output of the VFO will load it slightly, so it's not possible to precisely set it to a frequency using that method. However, you can use the scope to trim any adjustments in your circuit to see when the 'scope shows a signal that isn't weak and collapsing (such circuits will oscillate well in a range, and the output may collapse at extremes).
It probably won't look like an ideal sine-wave, this is OK. If you really want a voltage measurement, then even a peak-to-peak measurement (using cursors or automated measurements, either will be fine, there's no need for accuracy) is sufficient (I've not checked your documentation to see if p-p is what is being asked). If (say) the documentation says to adjust to 500mV p-p, then if it is 450mV or 550mV makes little difference.
If the measurement is very low, it may well collapse and produce unstable results. But there's a very good chance the unstable results are from a triggering issue. A screenshot/capture from your 'scope will help to narrow that down.
Once you're in the ballpark frequency based on what you've measured on the 'scope (automated measurements, but do a sanity check using cursors too, or with calculator and eyeballing it), and once the amplitude is in the ballpark you are expecting, then you could remove the 'scope, and then try to tune in your receiver to a known frequency transmission, and then fine-tune based on what you hear.
The screenshots below show two example 'scope traces for a particular VFO (different to your design, i.e. not related to that attached schematic) set to a center frequency, and to an extreme frequency.
Notice how the right-side screenshot looks more like a sinewave. However, that right-side screenshot actually shows a lower amplitude (see the volts per div setting on the screenshot), and and it is unstable (see the fuzziness on the trace).
The left side screenshot is actually the better setting. There, the amplitude is higher, and the signal is stable. The actual shape of it is less important, because it is normal for there to be harmonics that will cause that misshapeness.