Q : Can the tools of mechanism design be applied when the set of feasible decisions $D$ depends on the types $\theta$?
For example, when $d \in D$ represents the total funding allocated toward some good and $\theta_i$ represents the funds $i$ has available, $D = \{ d | 0 < d < \sum\limits_i\theta_i\}$.
Q : If not, is there some other approach that analyzes this sort of scenario?
Yes, they can, because the utility function typically depends on the type. A particular action unavailable to a type can be modelled as that action yielding so low utility that it is never chosen no matter what other players do. Then apply standard mechanism design to the appropriately modified utility function.
I agree with Sander's reply, but want to add that you have to find such a way to circumvent your issue. Standard mechanism design applies to Bayesian games. In a Bayesian game, the action space is type-independent, and since the designer does not know the types and can only decide on outcomes contingent on the agents' actions, we need to specify when the designer chooses an outcome that is "not allowed."
For example, such a problem shows up in mechanism design with evidence. It is discussed in this working paper by Roland Strausz (see footnote 1, this paper is superseded by another paper called "Principled Mechanism Design with Evidence" that I couldn't find online). The paper I linked seems to take Sander's approach (see the utility function on page 8), but also discusses what the other papers taking other approaches do.
