Here is one solution using a custom function and the incredibly versatile gtsummary::add_stat() function:

library(gtsummary)

# Create custom function using gtsummary syntax to be used in add_stat(). 
# `variable` will represent all the columns in your dataframe passed to `tbl_summary()`.
# Hardcode the variable against which you want corr (in this case `mpg`)

fn_add_tau_and_p <- function(data, variable, ...) {
  t <- cor.test(data[[variable]], data[["mpg"]], method = "kendall")
  data.frame(`tau` = style_sigfig(t$estimate), `p` = style_pvalue(t$p.value))
  
}

# test individually
fn_add_tau_and_p(mtcars, "hp")
#>       tau      p
#> tau -0.74 <0.001

# Now use `add_stat()` to add your function to `tbl_summary()`

mtcars |>
  select(hp, disp, wt, mpg, gear) |>
  tbl_summary(
  ) |>
  add_stat(fns = all_continuous() ~ fn_add_tau_and_p)

Note: you can also use broom::tidy() within your custom function as an alternative, as per example 2 here.