How plot bars on top of grid lines when using barplot?

I think this might fall inside:
"panel.first: an ‘expression’ to be evaluated after the plot axes are set up but before any plotting takes place. This can be useful for drawing background grids or scatterplot smooths. Note that this works by lazy evaluation: passing this argument from other plot methods may well not work since it may be evaluated too early"

So, probably, the best way to do it is draw the barplot in 3 steps:

# Draw the barplot
bp <- barplot(WRE, xaxt = 'n',xlab = '',yaxt = 'n',las = 3, width =0.5,space = 1.5, main=paste("How You Compare", sep=""), ylab="",  names.arg=NA,  col=c("red","blue"), cex.lab=1)
# add the grid
grid(nx=NA, ny=NULL)
# redraw the bars...
barplot(WRE, xaxt = 'n',xlab = '',yaxt = 'n',las = 3, width =0.5,space = 1.5, ylab="",  names.arg=NA,  col=c("red","blue"), cex.lab=1, add=T)
# Then you can add the axes
axis(1, at=bp, labels=c("Average Claims", "Your Claims"), tick=FALSE, las=1, line=-1, cex.axis=1) 
axis(2, at=axTicks(2), sprintf("$%s", formatC(axTicks(2), digits=9, big.mark=",", width=-1)), cex.axis=0.9, las=2)

You can try adding the grid to the plot after you set the axis, ie remove the panel.first argument and add grid(nx=NA, ny=NULL) after you add the axis