In classical field theory, the stress-energy tensor can be defined in terms of the variation of the action with respect to the metric field, or with respect to a frame field if spinors are involved. Of course, this assumes that the field theory is expressed in terms of an arbitrary metric (or frame) field, even if it's only a background field, so that we can define the variation. This is nicely reviewed in "Currents and the Energy-Momentum Tensor in Classical Field Theory — A fresh look at an Old Problem" (https://arxiv.org/abs/hep-th/0307199).
A similar definition works in quantum field theory, too, ignoring issues of regularization.
We can also have a non-lagrangian field theory whose equations of motion are not necessarily derivable from any lagrangian. We can still express the equations in terms of an arbitrary background metric (or frame) field, but how do we define the stress-energy tensor in this case? Do non-lagrangian field theories still have a stress-energy tensor?
I understand that Noether's theorem assumes a lagrangian, but the stress-energy tensor seems more fundamental, because even non-lagrangian systems couple to gravity, right? Shouldn't that mean that they have a stress-energy tensor?
The existence of a stress-energy tensor seems to be a standard axiom in non-lagrangian formulations of conformal field theory, but I don't know if these theories could be formulated in terms of a lagrangian, or if they are truly non-lagrangianable. (Can we please make that a word?)
Other posts about non-lagrangian(able) theories: