In my experience, many law firms want to see your law school transcript at the outset. It is often a prerequisite to even being considered. For these firms, the very short answer is "No, you can't send just your resume if you want your resume to be looked at." More to the point, though, you almost always want to send your transcript when applying to a firm.
How much weight a law firm gives to the transcript totally depends on the firm. Some law firms have strict grading criteria even when they consider more senior attorneys. Some firms will not consider any transcripts that have anything lower than a B. Some only want to consider lawyers who graduated at or near the top of their class. Bear in mind that although many firms want to see (or require that they see) your transcript, it is a smaller percentage of firms that are as strict about grades as the firms I just mentioned.