How to move y-axis labels away from R plot using lapply in R

You can define the ylab separately, like what you're doing for the xlab, and set the line parameter to define its distance from the plot (as stated in this post).

I got a running example from combining your code and @rawr's from your previous question.

set.seed(1)
z <- ts(matrix(rt(200 * 10, df = 3), 200, 10), start = c(1961, 1), frequency = 12) 
z <- z * 1e5 # to make "wide" y-axis labels

## vectors of x, y, and main labels
xl <- sprintf('x label %s', 1:10)
yl <- sprintf('y label %s', 1:10)
ml <- sprintf('main label %s', 1:10)


labes1 <- c("P(LNG)","","Volume(LNG)","","P(oil)","","Can.GDP","","US GDP","")
titles <- c("Levels","","","","","Log Difference","","","","")

par(mfrow = c(5, 2), mar = c(0.3, 6, 0, 2), oma = c(5, 0, 3, 2))
lapply(1:10, function(ii) {
  x <- z[, ii, drop = FALSE]
  plot(x, xlab = "Quarter", ylab = "", axes = FALSE) # set ylab to ""
  axis(2, las = 1)
  title(ylab = labes1[ii], line = 4) # set the line at an appropriate distance 
  box()
  if (ii %in% 9:10) {
    axis(1)
    title(xlab = 'Quarter', xpd = NA)
  }
  if (ii %in% 1:2)
    title(main = c('Levels', 'Log Difference')[ii], xpd = NA, line = 1)
})

The code above outputs the following graph for line = 4 :

and this plot for line = 3 :