Is it possible to replace the \mathbbm letters with less grainy ones?

This is just to honorably mention the dsfont package.

\documentclass{article}
\usepackage{bbm}
\usepackage{dsfont}
\usepackage{amsfonts}
\begin{document}
\begin{tabular}{lll}
\verb|\usepackage{bbm}| & \verb|\mathbbm| & $\mathbbm{A}$ \\ 
\verb|\usepackage{dsfont}| & \verb|\mathds| & $\mathds{A}$ \\ 
\verb|\usepackage{amsfonts}| & \verb|\mathbb| & $\mathbb{A}$ \\
\end{tabular}
\end{document}

the bbm fonts are bitmap fonts generated by MetaFont. If other double struck fonts are suitable many are available in scalable formats eg \mathbb font from the AMS but if you really want \mathbb then

Your log will show:

  .texlive2019/texmf-var/fonts/pk/ljfour/public/bbm/bbm10.600pk

That is a 600dp bitmap tweaked for an HP laserjet 4 printer (which you are probably not using:-)

If you modify the file to

\documentclass{report}
\usepackage{bbm}
\pdfpkresolution=1200
\begin{document}
$\mathbbm{A}$
\end{document}

The first time you run it metafont will fire up and make a higher resolution bitmap

 .texlive2019/texmf-var/fonts/pk/ultre/public/bbm/bbm10.1200pk

You can only choose resolutions that are in modes.mf

/usr/local/texlive/2019/texmf-dist/metafont/misc/modes.mf

so if you need really high resolution you may need to supply an additional mode definition.

You can load many different \mathbb alphabets using the mathalpha package, or if you can use LuaLaTeX or XeLaTeX, unicode-math. All are outline, rather than bitmap, fonts.

After loading the package and your fonts of choice, write \mathbb{A}.