If I understand your use case correctly, you are looking for a field on the job orders you are pulling over that will contain a link for a candidate to apply to that particular position, correct?
As Dave mentioned, there is not a "built in" web application process in Bullhorn which is why you are unable to see any type of URL in the api fields on those orders. The url would depend on how your client is currently accepting applications. It could be the open source job board Bullhorn posted to GitHub, or it could be a custom solution similar to the URL Dave provided.
Your client could use a custom field on the job order to store that url, however that creates some additional complexity since you then have to have a process to fill the field in. If the root application url ever changed, you would have to go back in and update any existing posted orders.
What I would recommend is using some sort of standard application url based on the job id. For example, something like https://company.com/careers/apply/12345
. The link would take candidates to a form where they can apply for job id 12345. That way all your application needs is the job id from each job and it can construct the url on the fly. If your client ever changes their root application url, all you need to do is make a quick update to your app (or better yet store the root as a configuration variable) and you are all set.