line break in tabularx environment

One possible solution is makecell from makecell package.

OFFTOPIC: In my opinion you should try to reduce the space of PCR and Template column to fit the whole primer sequence (most of it) on one line or at least give the orientation (5´-3´ end).

documentclass{article}
usepackage{tabularx,seqsplit}
usepackage{multirow}
usepackage{makecell}
usepackage{booktabs}

begin{document}

renewcommand{arraystretch}{2}

begin{table}[h!]
    centering
begin{tabularx}{textwidth}{@{} llX @{}}
hline
     textbf{PCR} & textbf{Template} & textbf{Primers} 
     hline
     textit{#1} & 355. Pdat-1+intron &        begin{tabular}{@{}l@{}}
                   makecell[l]{GCCTATTCCAGTATGACCCCTTTGAA}
                   addlinespace
                   makecell[l]{AGATACGGGTTTCACTTTTTCTACCGGTACCCTCCAAGG}
                 end{tabular}
    hline
     textit{#2} & 399.Pdat-1::SEC-24.2::mKate::UTR & begin{tabular}{@{}l@{}}
                   makecell[l]{TACCGGTAGAAAAAGTGAAACCCGTATCTCAAATCCCAAATGG}
                   addlinespace
                   makecell[l]{TTTGGATCAACATGAGCATTCGGATTCGC}
                 end{tabular} 
                 
         hline
         textit{Fusion} & Products from PCR#1 and #2 & begin{tabular}{@{}l@{}}
                   makecell[l]{GCCTATTCCAGTATGACCCCTTTGAA} 
                   addlinespace
                   makecell[l]{TTTGGATCAACATGAGCATTCGGATTCGC} 
                 end{tabular} 
                 
         hline
end{tabularx}
caption{Primers}
label{tab:Primers_CSKD}
end{table}

end{document}