Ok, I think I found the solution; there were a couple of issues. One is that my test case was looking for the case when 10-dollar strike Call on a underlying at 11-dollars would cost exactly 1 dollar --> no implied volatility premium, which is when volatility = 0. Not a good test case. When I correct for this, and try a few different methods, the solution works when we use the "Brent" Method":

CallImpliedVol[s_,k_,r_,t_,price_] := \[Sigma] /. FindRoot[FinancialDerivative[{"European", "Call"},
{"StrikePrice"->k, "Expiration"->t},{"Volatility" ->\[Sigma],"InterestRate"->r,"CurrentPrice"->s}]==price,
{\[Sigma],.01,10}, Method->"Brent"]

CallImpliedVol[11,10,.05,1,1.93]

Returns

FinancialDerivative::checknumeric: Parameters {Volatility??} cannot have non-numeric values.

Out[124]= 0.249851