grep json value of a key name. (busybox without option -P)

I found tons of threads who discussed "how to grep json values". But unfortunately useless for me and all who using grep from busybox (embedded linux). This grep version doesn't have the option "-P" (perl exp). Only "-E" (Extended Regexp) is available.

BusyBox v1.20.2 () multi-call binary.

Usage: grep [-HhnlLoqvsriwFE] [-m N] [-A/B/C N] PATTERN/-e PATTERN.../-f FILE [FILE]...

Search for PATTERN in FILEs (or stdin)

        -H      Add 'filename:' prefix
        -h      Do not add 'filename:' prefix
        -n      Add 'line_no:' prefix
        -l      Show only names of files that match
        -L      Show only names of files that don't match
        -c      Show only count of matching lines
        -o      Show only the matching part of line
        -q      Quiet. Return 0 if PATTERN is found, 1 otherwise
        -v      Select non-matching lines
        -s      Suppress open and read errors
        -r      Recurse
        -i      Ignore case
        -w      Match whole words only
        -x      Match whole lines only
        -F      PATTERN is a literal (not regexp)
        -E      PATTERN is an extended regexp
        -m N    Match up to N times per file
        -A N    Print N lines of trailing context
        -B N    Print N lines of leading context
        -C N    Same as '-A N -B N'
        -e PTRN Pattern to match
        -f FILE Read pattern from file

I have a json example:

{
  "one": "apple",
  "two": "banana"
}

Now, I want to extract the value e.g. "apple" from key "one".

grep -E '".*?"' file.json

Just an example how it should look like.

And btw: How to access groups from regex?

I would be grateful for any help or alternatives.