Replacing missing values

One dplyr and tidyr possibility could be:

df %>%
 group_by(quarter = substr(Period, 5, 6)) %>%
 mutate(Sales_temp = replace_na(Sales, last(na.omit(Sales)))) %>%
 group_by(quarter, na = is.na(Sales)) %>%
 mutate(constant = 1.05,
        Sales_temp = Sales_temp * cumprod(constant),
        Sales = coalesce(Sales, Sales_temp)) %>%
 ungroup() %>%
 select(1:2)

   Period Sales
   <chr>  <dbl>
 1 1999Q1  353.
 2 1999Q2  426.
 3 1999Q3  358.
 4 1999Q4  364.
 5 2000Q1  303.
 6 2000Q2  394.
 7 2000Q3  435.
 8 2000Q4  388.
 9 2001Q1  318.
10 2001Q2  414.
11 2001Q3  457.
12 2001Q4  407.
13 2002Q1  334.
14 2002Q2  435.
15 2002Q3  480.
16 2002Q4  428.
17 2003Q1  351.
18 2003Q2  456.
19 2003Q3  504.
20 2003Q4  449.

Or with just dplyr:

df %>%
 group_by(quarter = substr(Period, 5, 6)) %>%
 mutate(Sales_temp = if_else(is.na(Sales), last(na.omit(Sales)), Sales)) %>%
 group_by(quarter, na = is.na(Sales)) %>%
 mutate(constant = 1.05,
        Sales_temp = Sales_temp * cumprod(constant),
        Sales = coalesce(Sales, Sales_temp)) %>%
 ungroup() %>%
 select(1:2)