There is no interface to import the exact bibliography of another document. But in certain situations work-arounds are available.
All the information for citations and the bibliography lives in the .bbl file. That file is written by Biber following requests by biblatex in the .bcf file. So you can try to import the .bbl file of the other document to obtain its bibliography. This works as long as the set-up in both documents is simple: You can't have multiple refsections, the importing document can't cite sources that were not cited in the original document, etc.
Assuming you have a document called maindoc.tex with citations and bibliography that has gone through the complete compilation sequence of LaTeX, Biber , LaTeX, LaTeX you can then import the bibliography from maindoc (it lives in maindoc.bbl so that file must exist) in another document (coverletter.tex) with
\documentclass[american]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=authoryear, backend=biber]{biblatex}
\makeatletter
\newcommand*{\importbibfrom}[1]{%
\def\blx@bblfile{%
\blx@secinit
\begingroup
\blx@bblstart
\InputIfFileExists{#1.bbl}
{\blx@info@noline{... file '#1.bbl' found}%
\global\toggletrue{blx@bbldone}%
\blx@generate@bbl@mdfivesum@found{#1.bbl}}
{\blx@info@noline{... file '#1.bbl' not found}%
\typeout{No file #1.bbl.}%
\blx@generate@bbl@mdfivesum@notfound}%
\blx@bblend
\endgroup
% global sorting as this is called at BeginDocument
\csnumgdef{blx@labelnumber@\the\c@refsection}{0}}}
\let\blx@rerun@biber\relax
\makeatother
\importbibfrom{maindoc}
\begin{document}
\cite{sigfridsson}
\printbibliography
\end{document}
You only need to compile the importing document coverletter with LaTeX (at least twice), don't run Biber on that file.
You can cite all entries that are in the bibliography and get the exact same bibliography as maindoc.tex.
Assuming maindoc.tex was
\documentclass[american]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=authoryear, backend=biber]{biblatex}
\addbibresource{biblatex-examples.bib}
\begin{document}
\cite{sigfridsson,vizedom:related}
\printbibliography
\end{document}
the coverletter.tex from above yields

edit: See How to use refsection and xcite together? for guidance on how to use different refsections and for help with defernumbers.