Docker Compose: executable file not found in $PATH: unknown

As pointed out in @derpirscher's comment and mine, one of the issues was the permission of your script(s) and the way they should be called as the ENTRYPOINT (not CMD).

Consider this alternative code for your Dockerfile :

FROM node:16

WORKDIR /usr/src/app

COPY package*.json ./
COPY wait-for-it.sh ./
COPY docker-deploy.sh ./

# Use a single RUN command to avoid creating multiple RUN layers
RUN chmod +x wait-for-it.sh \
  && chmod +x docker-deploy.sh \
  && npm install --legacy-peer-deps

COPY . .

RUN npm run build

ENTRYPOINT ["./docker-deploy.sh"]

docker-deploy.sh script :

#!/bin/sh

# call wait-for-it with args and then start node if it succeeds
exec ./wait-for-it.sh -h "${DB_HOST}" -p "${DB_PORT}" -t 300 -s -- node start

See this other SO question for more context on the need for the exec builtin in a Docker shell entrypoint.

Also, note that the fact this exec ... command line is written inside a shell script (not directly in an ENTRYPOINT / CMD exec form) is a key ingredient for using the parameter expansion.
In other words: in the revision 2 of your question, the "${DB_HOST}:${DB_PORT}" argument was understood literally because no shell interpolation occurs in an ENTRYPOINT / CMD exec form.

Regarding the docker-compose.yml :

# version: '3.7'
# In the Docker Compose specification, "version:" is now deprecated.

services:
  my-service:
    build: .
    # Add "image:" for readability
    image: some-optional-fresh-tag-name
    # Pass environment values to the entrypoint
    environment:
      DB_HOST: postgres
      DB_PORT: ${DB_PORT}
      # etc.
    # Add network spec to make it explicit what services can communicate together
    networks:
      - postgres-network
    # Add "depends_on:" to improve "docker-run scheduling":
    depends_on:
      - postgres

  postgres:
    # container_name: postgres # unneeded
    image: postgres:14.3
    environment:
      POSTGRES_PASSWORD: ${DB_PASSWORD}
      POSTGRES_USER: ${DB_USER}
      POSTGRES_DB: my-service-db
      PG_DATA: /var/lib/postgresql2/data
    volumes:
      - pgdata:/var/lib/postgresql2/data
    networks:
      - postgres-network
    # ports:
    #   - ${DB_PORT}:${DB_PORT}
    # Rather remove this line in prod, which is a typical weakness, see (§)

networks:
  postgres-network:
    driver: bridge

volumes:
  pgdata:
    # let's be more explicit
    driver: local

Note that in this Docker setting, the wait-for-it host should be postgres (the Docker service name of your database), not 0.0.0.0 nor localhost. Because the wait-for-it script acts as a client that tries to connect to the specified web service in the ambient docker-compose network.

For a bit more details on the difference between 0.0.0.0 (a server-side, catch-all special IP) and localhost in a Docker context, see e.g. this other SO answer of mine.

(§): last but not least, the ports: [ "${DB_PORT}:${DB_PORT}" ] lines should rather be removed because they are not necessary for the Compose services to communicate (the services just need to belong to a common Compose network and use the other Compose services' hostname), while exposing one such port directly on the host increases the attack surface.

Last but not least:

To follow-up this comment of mine, suggesting to run ls -l docker-deploy.sh; file docker-deploy.sh in your myapp/ directory as a debugging step (BTW: feel free to do this later on then comment for the record):

Assuming there might be an unexpected bug in Docker similar to this one as pointed by @Lety:

I'd suggest to just replacing (in the Dockerfile) the line

RUN chmod +x wait-for-it.sh \
  && chmod +x docker-deploy.sh \
  && npm install --legacy-peer-deps

with

RUN npm install --legacy-peer-deps

and running directly in a terminal on the host machine:

cd myapp/
chmod -v 755 docker-deploy.sh
chmod -v 755 wait-for-it.sh

docker-compose --env-file .env up --build

If this does not work, here is another useful information you may want to provide: what is your OS, and what is your Docker package name? (e.g. docker-ce or podman…)