Our website uses WordPress but the developers are having issues integrating CiviCRM.
Should we go elsewhere? We want integration with our website but can we do this hosting CiviCRM outside of the website?
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It depends on what type of experience / skills they have. There are a lot of Wordpress 'developers' that wouldn't be able to go fully into a web development role.
However they likely have no experience of CiviCRM so they will have to learn a new solution from scratch so there will be issues along the way.
If they are having issues they can reach out to the community for advise on how to bring it forward and resolve any issues they are having.
Your title doesn't quite match the detail of the question. As Jade says it would help to have more detail, but answering the title question, I'd say that its an integration issue rather than an actual WordPress problem (many people run CiviCRM on WordPress successfully). So I'd say the website company needs to address the issues.
Looking at the more detailed question, it probably would be possible to host CiviCRM on WordPress elsewhere, but that isn't likely to make life easier. Your website developers are still going to have to integrate it into their site and you are also adding complexity by introducing a third party.
the answer may depend on how much you want to keep the WP developers involved in your WP site. Seem like several solutions eg