Yes, it can. We can determine the distance to the source if we have an idea of what's causing that shift in velocity.
Let's say that we have a source moving at a speed $v$ away from us. If it emits a photon of wavelength $\lambda_e$, we will observe that photon to have a wavelength of
due to the Doppler effect. This in turn means that we can calculate the object's speed if we can measure $\lambda_o$ and know $\lambda_e$:
Let's take a look at how this is used by radio astronomers.
One of the most important spectral lines at radio wavelengths is the 21 cm line, so named because it has a rest wavelength of 21 cm. It arises from a change in the spin of an electron in an atom of neutral hydrogen, which we sometimes call the spin-flip transition. The probability of such a transition is extremely low, but if you put together enough neutral hydrogen - say, in the form of an interstellar cloud - you can produce a detectable signal.
Astronomers have used the 21 cm line to map the Milky Way. We have a rough idea of how clouds should move throughout the galaxy. If we point our telescope at a particular cloud in the sky, we can measure its radial velocity via the 21 cm line and insert that velocity into models of the Milky Way's rotation to determine the cloud's location. This in turn allows us to figure out its distance to Earth. (Unfortunately, for most clouds of neutral hydrogen in the sky, each possible radial velocity yields two possible locations in the galaxy, so we have to break that degeneracy.)
The second way we can use radio waves to determine the distance to a source is by observing distant galaxies. On scales larger than galaxy clusters, the universe is expanding at an accelerating rate. Hubble's law tells us that a galaxy a distance $d$ from us should recede at a speed
where $H_0$ is Hubble's constant. If we have $v$, we can easily calculate $d$ from Hubble's law.
So, how do we determine the radial velocity? Well, we can certainly use the 21 cm line. Alternatively, if the source is a radio galaxy - a galaxy whose central supermassive black hole is causing intense radio emission - we can try to fit the galaxy's spectral energy distribution (SED), which shows how emission varies across different frequencies. If we have a model SED for the particular type of galaxy, we can fit its parameters to the observations. This will tell us, among other things, the redshift, and by extension the radial velocity.
Fitting SEDs isn't always the best method of measuring the radial velocity of high-redshift objects, and for both theoretical and instrumental reasons, we actually often use optical or infrared telescopes to study the galaxy's spectral lines, and use those to determine the redshift, rather than the SED. You can find a lot of papers talking about determining redshifts of radio galaxies - but by observing with visible light!