Yes, it is possible using 'conditional cell data' feature. Little modification required in lib jqgrid_dist.php

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$row[$col_name] = ( $r ? $c["default"] : '');
and replace with
$row[$col_name] = ( $r ? $c["default"] : $row[$col_name]);

Now, in column definition…

$col = array();
$col["title"] = "Company";
$col["name"] = "company";
$col["width"] = "30";
$col["align"] = "center";

$buttons_html = "<a target='_blank' href={website}'>{company}</a>"; // where website and company both are cols of grid.
$col["default"] = $buttons_html;

// use single quote for condition, and $row will have all columns data, to use in condition
$col["condition"] = '$row["website"] != ""';
$cols[] = $col;

$buttons_html = "<a target='_blank' href={website}'>{company}</a>";
should be
$buttons_html = "<a target='_blank' href='{website}'>{company}</a>";

… but even so, there are 2 issues:

1. The company field is url encoded (showing + instead of spaces). I have tried to fix this both by using the built in php urldecode function and str_replace("+"," ","{company}") but neither removes the +.

2. The website link is part of the data returned via $g-select_command = $SQL; but I have not put it in the grid. It does work (sort of) when I add it to the grid, but I have not worked out how to hide the column (as I don't want it displayed).
The other problem is that the link appears like this:
http://www.mywebsite.com/http%3A%2F%2Fwww.storewebsite.com

For html encoded data, use this formatter for company and website.
$col["formatter"] = "function(cellval,options,rowdata){return $.jgrid.htmlEncode(cellval);}";

Youcan hide website column using
$col["hidden"] = true;