You would need to provide many more specifics to give close to a yes or no answer here. Standing is total BS. As John Roberts said, standing is "a lawyer's game." For standing, the attorney needs to create a good package of injuries: in other words, an artfully worded complaint. Then you need a sympathetic judge. Finally, you would need to have a specific action that can be challenged. A "policy" would not work but you could challenge specific actions under a policy. You'd need to find a way to avoid a "generalized grievance."
In such an artful complaint, the attorney would have to be careful as to what is pled and what is not pled. If you do a Sherif Joe and put a statement like "I just want to get the government to enforce the law" before the court, you have lost on standing right there.
So, for example, DACA is called "guidance" by DHS (id est, a policy). However, there are specific actions under that policy that could be challenged. E.g., granting work permits.
Then there are very good injuries that could be artfully plead that might apply to a legal immigrant, such as increased competition, deprivation of procedural rights, and deprivation of statutory protections.
In short, the answer depends upon whether you can gather the facts to "package" an injury and the judge you get.
I'd cite cases but you don't have specific facts to make such a standing package. A good package maps the facts of the case to several established injuries.