There's a way to extract information from reports in JMP using a combination of using the GUI to investigate the report structure and scripting to pull specific information.
Step 1 is to do your analysis, then right click on the header closest to the information you want (red space below), then select Edit...Show Tree Structure.
In the window that pops up, find the info you're looking for, then the name of the header just above it in the hierarchy. In this case, NumberColBox(8) has the Odds Ratio, so you'll need to remember NumberColBox(8) for the scripting step.
Step 2 is to write the script. The first few lines in the script below are just for opening a file that this will work on; the magic starts at the "Export the report layer" comment. Basically, you're creating a separate object that has a copy of all the hierarchy of the report in it, then querying that separate object to get the text/number you need. In the script below, I send the odds ratio to the clipboard, since you mentioned needing to paste it. You could also use scripting to do other things with the value.
Names Default To Here( 1 );
//open the data file and run analysis, including odds ratio
dt = Open( "$SAMPLE_DATA/Lung Cancer.jmp" );
obj = dt << Contingency( Y( :Lung Cancer ), X( :Smoker ), Freq( :Count ), Odds Ratio( 1 ) );
//export the report layer, so it can be queried
r = obj << report;
//from looking through the "Show Tree Structure" manually to find the number needed
oddsratio = r[NumberColBox(8)] << Get; //pulls a list of what in that NumberColBox
show(oddsratio[1]); //shows the first item in the list (the odds ratio)
Set Clipboard (round(oddsratio[1],3)); //copies rounded odds ratio to the clipboard for easy pasting
Here's a link to a great resource for learning more about scripting the report layer.