The gt package allows for a few cell padding parameters to be passed within tab_options(). For example, data_row.padding and footnotes.padding. However, there are no parameters available to control the Table Header or Column Labels parts of the gt Table.
reprex with massive cell padding for demonstration:
gt_tbl <- gt(data = mtcars) %>%
tab_header(
title = "mtcars dataset") %>%
tab_options(data_row.padding = px(35))
gt_tbl