I'd like to write a JSON file using BASH but it seem's not working well..
My code :
sudo echo -e "Name of your app?\n"
sudo read appname
sudo cat "{apps:[{name:\"${appname}\",script:\"./cms/bin/www\",watch:false}]}" > process.json
Issue : -bash: process.json: Permission denied
Generally speaking, don't do this. Use a tool that already knows how to quote values correctly, like jq:
jq -n --arg appname "$appname" '{apps: [ {name: $appname, script: "./cms/bin/www", watch: false}]}' > process.json
That said, your immediate issues is that sudo only applies the command, not the redirection. One workaround is to use tee to write to the file instead.
echo '{...}' | sudo tee process.json > /dev/null
false}]} should be replaced with standard tick to work properly: jq -n --arg appname "$appname" '{apps: [ {name: $appname, script: "./cms/bin/www", watch: false}]}' > process.json 
To output text, use echo rather than cat (which outputs data from files or streams).
Aside from that, you will also have to escape the double-quotes inside your text if you want them to appear in the result.
echo -e "Name of your app?\n"
read appname
echo "{apps:[{name:\"${appname}\",script:\"./cms/bin/www\",watch:false}]}" > process.json
If you need to process more than just a simple line, I second @chepner's suggestion to use a JSON tool such as jq.
Your -bash: process.json: Permission denied comes from the fact you cannot write to the process.json file. If the file does not exist, check that your user has write permissions on the directory. If it exists, check that your user has write permissions on the file.
sudo echoandsudo readon the first two lines look completely pointless to mesudo vim process.jsonit works...vimopens the file; with> process.json, the shell opens the file beforesudoactually runs.