How do I define norm delimiters?

You've been informed that norm is an "undefined control sequence". You further mention, in a comment, that you load the amsmath and amssymb packages. (Aside: Since the amssymb package loads the amsfonts package automatically, you don't need to load the amsfonts package explicitly.)

You have two main options:

1. Remove the two norm directives from the code. And, while you're at it, do remove the left and right sizing directives as well, as they don't do anything here except create code clutter.

   documentclass{article} % or some other suitable document class
   usepackage{amsmath,amssymb}
   usepackage{old-arrows} % optional (for smaller arrowheads) 
   
   begin{document}
   begin{equation}
       R = Vertoverrightarrow{q_1c}Vert = Vertoverrightarrow{q_2c}Vert
   end{equation}
   end{document}
   

2. Remove the leftVert and rightVert directives and define a macro called norm. I'd like to suggest you load the mathtools package -- a superset of the amsmath package -- for its DeclarePairedDelimiter macro to define norm.

   documentclass{article} % or some other suitable document class
   usepackage{mathtools,amssymb}
   DeclarePairedDelimiter{norm}{lVert}{rVert} % define a "norm" macro
   usepackage{old-arrows} % optional (for smaller arrowheads) 
   
   begin{document}
   begin{equation}
       R = norm{overrightarrow{q_1c}} = norm{overrightarrow{q_2c}}
   end{equation}
   end{document}
   

With both approaches, you'll get the following output:

Of the two approaches, the second is definitely more "LaTeX-y" as the code (here: norm{...}) emphasizes the meaning of what you're inputting. This conforms better to LaTeX's design philosophy of distinguishing as much as possible between higher-level meaning and lower-level typesetting aspects of the code.