Here is another tidyverse option, where I convert to date (and then reformat), then we can fill in the date, so that we can use that to group. Then, get the sum for each date.
library(tidyverse)
df %>% 
  mutate(permitnum = format(as.Date(permitnum, "%m/%d/%Y"), "%m/%d/%Y")) %>% 
  fill(permitnum, .direction = "down") %>% 
  group_by(permitnum) %>% 
  summarise(total_amount = sum(as.numeric(amount), na.rm = TRUE))
Output
  permitnum  total_amount
  <chr>             <dbl>
1 06/01/2022         70.4
2 06/02/2022         61.7
Data
df <- structure(list(permitnum = c("6/1/2022", "ascas", "olic", "6/2/2022", 
"avrey", "fev", "grey"), amount = c("na", "30.00", "40.41", "na", 
"17.32", "32.18", "12.20")), class = "data.frame", row.names = c(NA, 
-7L))