For this question, I've borrowed the system of equations presented in https://mathematica.stackexchange.com/questions/153346/solving-for-a-single-variable-across-multiple-equations/153351#153351.   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

[![enter image description here][1]][1]

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

[![enter image description here][2]][2]

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

[![enter image description here][3]][3]

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]]

[![enter image description here][4]][4]



This also doesn't give $\sigma \mathrm M$:
    
    Reduce[elim, \[Sigma]M] // Simplify

[![enter image description here][5]][5]

I'm using MMA 12.2 on MacOS 10.13.6:

[![enter image description here][6]][6]

Update
=
MichaelE2 noticed that Reduce does show the desired solution if Simplify is not applied: 

    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}] (*the missing solution is seen if Simplify is not applied*)
    sm = 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}][[3, 
       2]](*here I've pulled out the missing solution*)
    Simplify@sm
    (*when Simplify is applied to the missing solution, it says it's
    equal to zero*)

[![enter image description here][7]][7]


  [1]: https://i.sstatic.net/2uv9h.png
  [2]: https://i.sstatic.net/l27ko.png
  [3]: https://i.sstatic.net/iwdxo.png
  [4]: https://i.sstatic.net/Ycg3C.png
  [5]: https://i.sstatic.net/dwMpA.png
  [6]: https://i.sstatic.net/NduiS.png
  [7]: https://i.sstatic.net/9FqGi.png