Yes- VNA's measure not only reflection but also transmission.
The most common configuration for a VNA is in reflection mode- S11, etc. Reflected voltage/Incident Voltage measured with a single port. The other common operating mode is transmission mode (two port mode) where you measure the ratio of Transmitted voltage to Incident voltage - S21. The second port is a Receiver and acts similarly to a spectrum analyzer with a few wrinkles. S21 as displayed is the ratio of received voltage on port 2 to incident voltage on port 1 and this is a vector ratio (it accounts for phase). With your antenna, you won't connect port 1 to anything (terminating it would be best). The s21 ratio will be zero everywhere except where you get a a signal from the antenna. You'll get an amplitude that is greatest when the antenna output (s2 voltage) is greatest, ignore the phase, it will be meaningless- its not driven from the signal generator. You can derive how good a spectrum analyser this makes. Its only OK since you're not making a ratiometric measurment, there is a small IF offset, the dynamic range isn't great and bandwidth and resolution isn't what you would typically want but it will work. It will give you signal strength after a fashion. A spectrum analyzer like the Tiny SA would be a better tool for this. There may be a way to just readout S2 voltage on the Tiny VNA but its not in the menus- this would make it a bit more like an SA. Will the PC software allow this?- I don't know.