For this question, I've borrowed the system of equations presented in Solving for a single variable across multiple equations. I've converted all values to exact numbers, to provide both Solve and Reduce with exact input.

I found that Solve is able to solve this system for the requested variable, while Reduce is not. I'm curious why that is.

The following (undocumented) syntax means: "Solve the given system of equations for $\sigma \mathrm M$, while eliminating VF and Vs:

Solve[Log10[f/(1 - f)] == pH - pKa + F Vs/(23/10 R T) && \[Sigma]M == 
    cF VF && Ee - Epzc == VF + Vs, \[Sigma]M, {Vs, VF}] // Simplify

One can also obtain the above output with either of these approaches:

elim = Eliminate[
   Log10[f/(1 - f)] == pH - pKa + F Vs/(23/10 R T) && \[Sigma]M == 
     cF VF && Ee - Epzc == VF + Vs, {Vs, VF}];
Solve[elim, \[Sigma]M] // Simplify
Solve[Log10[f/(1 - f)] == pH - pKa + F Vs/(23/10 R T) && \[Sigma]M == 
    cF VF && Ee - Epzc == VF + Vs, \[Sigma]M, {Vs, VF}, 
  Method -> Reduce] // Simplify

By contrast, here's what I get with Reduce, using the same undocumented syntax:

Reduce[Log10[f/(1 - f)] == pH - pKa + F Vs/(23/10 R T) && \[Sigma]M ==
     cF VF && Ee - Epzc == VF + Vs, \[Sigma]M, {Vs, VF}] // Simplify

The issue doesn't appear to be the use of undocumented syntax, since Reduce doesn't solve for $\sigma \mathrm M$ using this standard syntax, while Solve does:

Solve[Log10[f/(1 - f)] == pH - pKa + F Vs/(23/10 R T) && \[Sigma]M == 
    cF VF && Ee - Epzc == VF + Vs, {\[Sigma]M, Vs, VF}] // Simplify
Reduce[Log10[f/(1 - f)] == pH - pKa + F Vs/(23/10 R T) && \[Sigma]M ==
         cF VF && Ee - Epzc == VF + Vs, {\[Sigma]M, Vs, VF}] // Simplify

Furthermore, Reduce does recognize the undocumented syntax:

Reduce[x + 2 y + 3 z == 4 b c && 3 x + 4 y + 5 z == 6 && 
   6 x + 7 y + 8 z + d == 9, x, {y, c}][[1, 1]]
Reduce[x + 2 y + 3 z == 4 b c && 3 x + 4 y + 5 z == 6 && 
   6 x + 7 y + 8 z + d == 9, y, {x, c}][[1, 1]]
Reduce[x + 2 y + 3 z == 4 b c && 3 x + 4 y + 5 z == 6 && 
   6 x + 7 y + 8 z + d == 9, c, {x, y}][[2, 2]]

This also doesn't give $\sigma \mathrm M$:

Reduce[elim, \[Sigma]M] // Simplify

I'm using MMA 12.2 on MacOS 10.13.6: