Question: Is it true that for a ring $\mathit{R}$, the flat left (right) $\mathit{R}$-modules form a reflective subcategory of the category of all left (right) $\mathit{R}$-modules if and only if $R$ is right (left) semihereditary?
It is easy to see that the flat $R$-modules form a reflective subcategory if $\mathit{R}$ is a Prüfer domain, because then flat $\mathit{R}$-modules would coincide with torsion-free ones, and the quotient of any $\mathit{R}$-module by its torsion submodule would then give a left adjoint to the inclusion of flat (= torsion-free) $\mathit{R}$-modules in $\mathit{R}$-mod. The statement also clearly holds if $\mathit{R}$ is von Neumann regular, because then every $\mathit{R}$-module would be flat, and any category is trivially a reflective subcategory of itself.
The equivalent conditions under the "Relation with semihereditary rings" section of the Wikipedia article on torsionless modules imply that for a left semihereditary ring $R$, the flat right $R$-modules are closed under arbitrary submodules (equivalent to the same condition for flat left $R$-modules) and direct products (equivalent to $R$ being left coherent), and hence might form a reflective subcategory of the category of all right $R$-modules. If true, then this would answer the question in the "if" direction, but the "only if" direction must also be answered, perhaps with a counterexample if there is one.
It uses slightly different language, but Proposition 2.1 of
Asensio Mayor, J.; Martínez Hernández, J., On flat and projective envelopes, J. Algebra 160, No. 2, 434-440 (1993). ZBL0802.16003.
shows (switching left/right conventions to fit this question) that flat left $R$-modules being a reflective subcategory is equivalent to $R$ being right coherent with weak global dimension at most $2$.
So it is true that being right semihereditary is a sufficient condition (since right semihereditary rings are right coherent and have weak global dimension at most $1$).
But it is not a necessary condition, as for example $\mathbb{Z}[x]$ is right coherent with weak global dimension $2$, but is not right semihereditary.