GitLab CI Vault Integration does not work as expected

Problem

Today I tried the GitLab CI Vault integration to replace manual commands as shown below. The gitlab-integration job fails. The manual jobs succeeds with the correct data shown. I get a permissio denied error, but the same data is requested with commands which work with the JWT as shown below. Do I need to add some information for the Vault Integration or is my .gitlab-ci.yml faulty?

Thanks for any help!

Error Message from gitlab-integration job

Resolving secrets
Resolving secret "cert"...
Using "vault" secret resolver...
ERROR: Job failed (system failure): resolving secrets: reading secret: reading from Vault: api error: status code 403: 1 error occurred:
	* permission denied

Output from working manual job

$ export VAULT_ADDR=https://vault.+++.de:443
$ export VAULT_TOKEN="$(vault write -field=token auth/jwt/login jwt=$CI_JOB_JWT)"
$ vault kv get certificates/+++.de
====== Metadata ======
Key              Value
---              -----
created_time     2020-12-06T19:11:21.705026445Z
deletion_time    n/a
destroyed        false
version          1
======= Data =======
Key            Value
---            -----
certificate    -----BEGIN CERTIFICATE-----
+++
-----END CERTIFICATE-----
key            -----BEGIN RSA PRIVATE KEY-----
+++
-----END RSA PRIVATE KEY-----

Example

gitlab-integration: # Does not work
  stage: build
  variables:
    VAULT_SERVER_URL: https://vault.+++.de:443
# With or without these variables, it does not change the output
#    VAULT_AUTH_PATH: jwt 
#    VAULT_AUTH_ROLE: gitlab-ci
  secrets:
    cert:
        vault: certificates/+++.de
  image: registry.+++.de/+++/images/cpi
  script:
    - cat $cert

manual:  # Works without problems
  stage: build
  image: registry.###.de/+++/images/cpi # Image with Vault installed
  script:
    - export VAULT_ADDR=https://vault.+++.de:443
    - export VAULT_TOKEN="$(vault write -field=token auth/jwt/login jwt=$CI_JOB_JWT)"
    - vault kv get certificates/+++.de

For privacy reasons, some values are removed and replaced with +++.