Core component of SQL Server for storing, processing, and securing data
Express to Evaluation is not a supported upgrade path. For one thing, Express is licensed for production use. Evaluation is not.
You could install a new instance that runs Evaluation, and attach the database to that instance. However, I get the impression that you are usage of the database is for production, so you would violate the license agreement for Evaluation.
So, yes, your company will need to buy you a license for Standard Edition for you. How quickly the license and the download becomes available is nothing I can answer, as those routines are up to the reseller. But I would expect it to be a swift operation that is sorted out in a day as soon as the payment is processed.