Is there a biblical basis for God calling one to place one's career above their other duties? I've tried to phrase this question several ways now, and none of them are really to my liking. I will just ask it clumsily and provide more details as they are asked of me.

Does God ever call people to lives devoted to their careers?

For example, suppose someone says to himself "God has called me to forego starting a family in favor of devoting my life to being the best dog groomer I can be." I choose a particularly strange career on purpose. The work of a doctor and the work of a dog groomer is equally meaningless if it is not blessed by God. I suppose my question, therefore, is whether there is precedent that God asks for and accepts such a life, or whether this hypothetical dog groomer is deluding himself.
There is no specific Biblical precedent for such a claim - at least not in your example. But there is also no Biblical reason to refute such a claim, either.  (Unless the claim is a direct contradiction of clear Biblical teaching.)

The Scriptures give examples of people called to be Prophets, Evangelists, Pastors, and Preachers (Ephesians 4:11) and other ministries that directly serve God, or God's purpose. He has also ordained Kings. (Saul and David, for example).

Nowhere in Scripture is anyone specifically called to be a baker, a server, or other secular job.

There is validity and Biblical precendent in a claim to a career serving God as a Missionary, for example, or a Pastor, Priest, or other Church official.  We are all called to serve God in one way or another, and most traditions believe that God specifically calls people into positions of authority, such as those mentioned previously.  Exactly how they are called, and how they can be sure is likely the subject for a different question, as it varies based on factors such as denomination, interpretation of Scripture, and tradition.

However...

The Bible does tell us that no matter what our career is, we should do it "as unto the Lord", meaning that we should do our best as if we were working for God Himself.  Several examples of those serving honorably doing ordinary (secular) tasks and jobs do abound.  The woman at the Well giving water, for example, or the people called to feed the hungry, fight in the battles, serve their masters.  Being a good servant is a basic Christian principle, and is both commanded and honored in Scripture.

To answer your comment, it is entirely possible that God does call us to serve in secular jobs.  God is, after all, sovereign.  God alone knows where the line is between His sovereignty, and the free will He allows us.