Risk-neutrality isn't really a property of a model. Instead, it describes a certain calibration of a model (almost always represented by an SDE).
We say a model has been calibrated to risk-neutral probabilities if
- model parameters can be inferred from traded security prices, and
- there's some anti-arbitrage assumption and hedging scheme available for those traded security prices
The machinery is frequently abused to say that a risk-compensated model (with a nontrivial term for market price of risk) has been calibrated risk-neutrally. That's mainly because the math is all the same. Note also that even in the absence of hedging arguments, one can still often make large-$N$ portfolio arguments for security prices to all be priced on a consistent calibration.
Now, in the specific case of the Vasicek model, we can imagine calibrating its parameters to, say, swap and swaption prices. We won't match them all, but we'll do our best. And to the extent the model represents reality, any new securities we see in the market can be fairly priced using the model.
If we were instead calibrating the Vasicek model to a time series of overnight rates, our calibration would not be risk-neutral, and would be more appropriate for risk computations.
Finally, note that the short rate in the Vasicek (or similar) model is not an investable security. Market price of risk would not be applied directly to it.