write_hex() {
     hex="$1"
     echo "$hex" | sed -e 's/../&\n/' | while read -r hexbyte; do
      printf "\\x$hexbyte"
     done
    }

read_int16 and read_int32 don't work on big-endian systems, or if int is 16 bits instead of 32. the latter issue can be easily fixed by explicitly specifying -td2/-td4, but the former issue is not so easy. I think it requires either figuring out the endianness beforehand, or better, something like this:

    od -An -tx1 -j"$offs" -N4 "$path" | while read a b c d; do
     echo $(((0x$a << 24) | (0x$b << 16) | (0x$c << 8) | 0x$d))
    done

'for x in $y; do printf "$a%s$b" "$x"; done' is equivalent to 'printf "$a%s$b" $y' (assuming neither a nor b contain format specifiers). similarly, 'for i in {1..100}; do printf "$a"; done' is equivalent to 'printf "$a%s.0" {1..100}'. unfortunately, brace expansion is not POSIX, but these are both significantly more efficient (both in code size and execution time) than the loop methods.

sha1sum is not POSIX. I think shell arithmetic provides you enough tools to implement https://en.wikipedia.org/wiki/SHA-1#SHA-1_pseudocode directly, although it may be slightly slower than a C implementation. awk is probably faster than shell.