I have a process running on Slackware 13.37 that outputs data to its terminal window. Is it possible to read/redirect this output to ssh window/PuTTY without terminating/restarting the process?
You can capture the output using shell redirection or via a program such as script -- provided that your program was started "in the usual way". (It is possible to write via special devices to other terminals, given appropriate permissions).
One assumes that you already know about redirecting output, e.g.,
foo >bar
but have some case, e.g., a GUI program which continues to write to the terminal.
Without worrying about interfering with a program by redirecting its output, I would run script and start the program within the shell that script starts. Then anything written from that shell would go to the typescript file (by default).
On the other hand, GUI programs which start a terminal window may/may not be configurable to allow customizing them with a startup script that can capture output.
As noted in How to open process again in linux terminal?, it is possible to attach to a running process with strace, given its process-ID. Using the -e option as described in 7 Strace Examples to Debug the Execution of a Program in Linux, you could just extract write calls.
Keep in mind with strace that nonprinting characters from the writes are converted to printable text, and that strace displays function arguments up to a fixed limit (which you can adjust using the -s option). The output of strace can be redirected (and it need not be run on the same terminal as the original process).