No, vega cannot be derived in a model-free manner. The reason for this is because in contrast to delta and gamma, there are multiple definitions of vega, and an even deeper underlying reason may be that spot volatility is not an observable. The consequence of this is that 'vega' starts to depend on the quoting mechanism; i.e. do you express the market price of an option in terms of a local vol model, a stochastic vol model, a jump diffusion, or Black Scholes with an implied volatility? All these different quoting mechanisms will lead to a different concept and value of 'vega'.
As an example; suppose you are able to fit a pure jump model to the market price of options for a particular time to maturity. What is vega in a pure jump model? However you can (always) translate your pure jump model prices to Black-Scholes prices with implied vols that depend on strike. Then you do have vega. Hope this example makes it clearer.
Regarding theta: here too there is no model-free quantity. Recall that in SV models the theta of an option balances the gamma, vega, vanna, volga. As only gamma potentially is model-free, you cannot hope to have model-free theta.
As regards delta and gamma: Here under some circumstances (for the class of so-called homogeneous stochastic vol models) there is a model free definition, and this is (partly) because the spot price is an observable and therefore has to occur in any function you use to quote the market price by.
Last remark: as option market prices can always be expressed in terms of Black-Scholes prices with strike dependent IV, BS vega is always well-defined.