Let $C^\infty(\mathbb{R})_{int}$ denote the subspace of $C^\infty(\mathbb{R})$ whose elements are integrable on $[0,\infty)$, and let $C^\infty(\mathbb{R})_{int}^0$ denote the codimension one subspace of functions whose integral is zero. Here's a rephrasing of the desired properties of $A$ and $T$:
The restriction of $T$ to integrable (on $\mathbb{R}_{\geq 0}$) smooth functions C^\infty(\mathbb{R})_{int}$ lands in a distinguished subspace $\mathbb{R} \subset A$, and is given by ordinary integration.
As far as I can tell, there are lots of subspaces of
[Edit:] Let $C^\infty(\mathbb{R})$ that are closed under addition X$ is a space of integrable smooth functions , and are closed under the action of addition by $C^\infty(\mathbb{R})_{int}$, such that $\mathbb{R} \rtimes \mathbb{R}^\times_{>0}$. Indeed, essentially any asymptotic growth class yields such a space. Taking mathbb{R}^\times_{>0}$ acts freely on the span of quotient vector space $X/C^\infty(\mathbb{R})_{int}$. If a set universal target $A$ for integration existed, then $X/C^\infty(\mathbb{R})_{int}^0$ should admit an injection to $A$, because your list of such spaces yields conditions specifies no new further relations, so . The problem (as pointed out by Tao) is that there are lots of smooth functions with nontrivial stabilizer in $\mathbb{R} \rtimes \mathbb{R}^\times_{>0}$.
I think a universal recipient common method to removing such a difficulty is to ignore the requirement that integration be $A$ \mathbb{R} \rtimes \mathbb{R}^\times_{>0}$-equivariant. Then your universal space is necessarily much larger than just $\mathbb{R}$.C^\infty(\mathbb{R})/C^\infty(\mathbb{R})_{int}^0$.
