What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?
In place of a detailed answer, let me point to Internal categories, anafunctors and localisations, but more specific to your case is the diffeological groupoids in Smooth loop stacks of differentiable stacks and gerbes.
To answer a more specific question here:
Precisely in the condition of essential surjectivity (ES) we need a notion of surjective submersion but I don't know the analogue of surjective submersion for generalized smooth spaces
For diffeological spaces, and I would imagine any generalised smooth spaces that can be considered as perhaps special sheaves on the category of manifolds, the type of map you want is subduction. I don't have a good canonical (nLab!) reference, but there some discussion in this answer, and such maps appear in Konrad Waldorf's work on gerbes. Subductions is also discussed (briefly) in the second linked paper above.
Apologies for the late answer, I wish I'd found this earlier!
My MSc thesis was actually mainly devoted to developing a notion of Morita equivalence for diffeological groupoids! I don't think I'll have much to add to the answers by David Roberts and Joel Villatoro, and diffeological groupoids are clearly more specific than what you're looking for, but for what it's worth I'd like to contribute my two cents. You can find my thesis here: Diffeology, Groupoids & Morita Equivalence. I also wrote a paper about the main result (a "Morita Theorem"), which is available on the arXiv here: arXiv:2007.09901 (and which has been accepted for publication in the Cahiers)!
The approach I focussed on in my thesis is that of biprincipal bibundles. This is a slightly different point of view than your question, but it turns out that this gives a notion of Morita equivalence that is equivalent (no pun intended) to the definition using weak equivalences (see Section 5.1.3 of the thesis). The paper by Meyer and Zhu, mentioned in one of David Roberts' comments, also uses this point of view. I don't know to what extent their theory can be used for diffeology. I mainly wanted to focus on this part of your question:
"...I don't know the analogue of surjective submersion for generalized smooth spaces."
As David Roberts mentions, one sensible option is to replace surjective submersions by subductions. The entire point of my thesis is essentially to show that the Lie groupoid theory still works if you do this. (If you're looking for a reference on subductions, Section 2.6 of my thesis discusses them in some detail.) This also leads us to an important point that Joel Villatoro raises:
"By the way, for my part I would recommend taking surjective local subductions as your submersion for the diffeological category."
The motivation for this could be that the local subductions between smooth manifolds are exactly the surjective submersions (this is proved in the Diffeology textbook by Iglesias-Zemmour), hence directly generalising surjective submersions to the diffeological setting (which is what we were after!). However, as just mentioned, choosing subductions still makes everything work. Besides this, here are two more reasons to consider choosing subductions over local subductions:
There are naturally occurring bundle-like objects in diffeology that are subductions, but not local subductions. The main examples I have in mind are the internal tangent bundles. For example, if you consider diffeological space that is the union of the two coordinate axes in $\mathbb{R}^2$, its internal tangent bundle is 2-dimensional at the origin, but 1-dimensional everywhere else (see Example 3.17 in arXiv:1411.5425 by Christensen and Wu). The internal tangent bundle of this space is then a subduction, but not a local subduction (thanks to an argument by Christensen). If we want to study such objects to occur in our theory of diffeological groupoid bundles and Morita equivalence, we need to allow subductions and not just local subductions.
The main reason that we assume the source and target maps of a Lie groupoid $G\rightrightarrows G_0$ are surjective submersions is to ensure that the fibred product $G\times_{G_0}G$ of composable arrows is again a smooth manifold. Since the category $\mathbf{Diffeol}$ of diffeological spaces is (co)complete, this assumption becomes redundant. However, the source and target maps of a diffeological groupoid are always subductions!
I elaborate on the choice of subductions over local ones in Sections 4.2 and 4.4.3 in my thesis. My answer to your main question in the setting of diffeology is therefore the same as that of David Roberts: an appropriate notion of Morita equivalence for diffeological groupoids is exactly as for Lie groupoids, but with surjective submersion replaced by subduction.
As to a more structured approach to generalising surjective submersions to the setting of generalised smooth spaces (your actual main question): I believe that the subductions in the category $\mathbf{Diffeol}$ of diffeological spaces are exactly the strong epimorphisms, cf. Proposition 37 in arXiv:0807.1704 by Baez and Hoffnung. In a more abstract setting of generalised smooth spaces you could therefore consider trying to use the strong epimorphisms to replace the surjective submersions!
I know this is a little late but I discuss this in the first two chapters of my thesis here:
https://arxiv.org/abs/1806.01939
Basically, as you mentioned, what you need is a notion of surjective submersion which generalizes surjective submersions of smooth manifolds. Once you have that, the definition falls out of it by the usual theory. In my thesis, I talk about the case where we are given a site, equipped with a distinguished set of morphisms which are the 'submersions'. That distinguished set of morphisms has to have a few properties which you can find in the definition of good site in the first chapter of my thesis.
The short version is that your category needs to be reasonably compatible with the grothendiek topology (i.e. morphisms are characterized by locally) and your notion of surjective submersions should generate the Grothendiek topology.
The other main property is that if you have a bunch of submersions $s_i \colon P_i \to B$ with images covering $B$ and some coherent transition maps, you should be able to glue the $P_i$ into a single submersion $P \to B$. Lastly, you need that if $f \circ g $ is a submersion then $f$ is a submersion.
The main difference between my thesis and the paper of Roberts and Vozzo is that they focus on when the category can be localized by a category of fractions method. My thesis is mainly concerned with constructing a 2-categorical equivalence between bibundles of internal groupoids and presentable sheaves of groupoids.
By the way, for my part I would recommend taking surjective local subductions as your submersion for the diffeological category. That's my two-cents anyway.