Answer by Kamil Maciorowski for Need to expand a variable in a HEREDOC that is in quotes

TL;DR

Use <<EOF instead of <<'EOF'.

This is what POSIX says about here-documents [emphasis mine]:

[n]<<word    here-documentdelimiter

[…]

If any part of word is quoted, the delimiter shall be formed by performing quote removal on word, and the here-document lines shall not be expanded. Otherwise, the delimiter shall be the word itself.

If no part of word is quoted, all lines of the here-document shall be expanded for parameter expansion, command substitution, and arithmetic expansion. […]

In your case word is 'EOF' (quoted!), so $package inside the here-document is not expanded. Drop these quotes and you will get the desired result.

cat > out.json <<EOF{"apps": [    {"cwd":"/usr/local/$package"    }}EOF

Note this has nothing to do with quotes that surround the variable inside the here-document.