How can I perform and store linear regression models between all continuous variables in a data frame?

Assuming you need pairwise comparisons between all columns of mtcars, you can use combn() function to find all pairwise comparisons (2), and perform all linear models with:

combinations <- combn(colnames(mtcars), 2)

forward <- list()
reverse <- list()

for(i in 1:ncol(combinations)){
  forward[[i]] <- lm(formula(paste0(combinations[,i][1], "~", combinations[,i][2])), data = mtcars)

  reverse[[i]] <- lm(formula(paste0(combinations[,i][2], "~", combinations[,i][1])), data = mtcars)
}

all <- c(forward, reverse)

all will be your list with all of the linear models together, with both forward and reverse directions of associations between the two variables.

If you want combinations between three variables, you can do combn(colnames(mtcars), 3), and so on.