Using gsub() in awk on the last ;-delimited field:
$ awk -F ';' 'BEGIN { OFS=FS } { gsub(" ", ",", $NF); print }' file
some name;another thing; random;,value,value,value,value,value
Using sed and assuming we'd like to replace all spaces after the last ; with commas:
$ sed 'h;s/.*;//;y/ /,/;x;s/;[^;]*$//;G;s/\n/;/' file
some name;another thing; random;,value,value,value,value,value
Annotated sed script:
h ; # Duplicate line into hold space
s/.*;// ; # Delete up to the last ;
y/ /,/ ; # Change space to comma in remaining data
x ; # Swap pattern and hold spaces
s/;[^;]*$// ; # Delete from the last ;
G ; # Append hold space delimited by newline
s/\n/;/ ; # Replace the embedded newline with ;
; # (implicit print)
The "hold space" is a separate storage buffer that sed provides. The "pattern space" is the buffer into which the data is read from the input and to which modifications may be applied.