Here's a version I implemented using Rcpp. I compared pmin with my version, and my version is roughly 3 times faster.
library(Rcpp)
cppFunction("
  NumericVector min_vec(NumericVector vec1, NumericVector vec2) {
    int n = vec1.size();
    if(n != vec2.size()) return 0;
    else {
      NumericVector out(n);
      for(int i = 0; i < n; i++) {
        out[i] = std::min(vec1[i], vec2[i]);
      }
      return out;
    }
  }
")
x1 <- rnorm(100000)
y1 <- rnorm(100000)
microbenchmark::microbenchmark(min_vec(x1, y1))
microbenchmark::microbenchmark(pmin(x1, y1))
x2 <- rnorm(500000)
y2 <- rnorm(500000)
microbenchmark::microbenchmark(min_vec(x2, y2))
microbenchmark::microbenchmark(pmin(x2, y2))
The microbenchmark function output for 100,000 elements is:
> microbenchmark::microbenchmark(min_vec(x1, y1))
Unit: microseconds
            expr     min       lq     mean  median       uq
 min_vec(x1, y1) 215.731 222.3705 230.7018 224.484 228.1115
     max neval
 284.631   100
> microbenchmark::microbenchmark(pmin(x1, y1))
Unit: microseconds
         expr     min       lq     mean  median      uq      max
 pmin(x1, y1) 891.486 904.7365 943.5884 922.899 954.873 1098.259
 neval
   100
And for 500,000 elements:
> microbenchmark::microbenchmark(min_vec(x2, y2))
Unit: milliseconds
            expr      min       lq     mean   median       uq
 min_vec(x2, y2) 1.493136 2.008122 2.109541 2.140318 2.300022
     max neval
 2.97674   100
> microbenchmark::microbenchmark(pmin(x2, y2))
Unit: milliseconds
         expr      min       lq     mean   median       uq
 pmin(x2, y2) 4.652925 5.146819 5.286951 5.264451 5.445638
      max neval
 6.639985   100
So you can see the Rcpp version is faster. 
You could make it better by adding some error checking in the function, for instance: check that both vectors are the same length, or that they are comparable (not character vs. numeric, or boolean vs. numeric).