Can I reapply to the same firm for the same job if I see it is still posted?

Answer:

If you reapply and your resume is reviewed, it just may be the case that your credentials appear stronger than if your application had been reviewed originally. After firms talk with their stellar applicants, investing time in the process as the need for additional help is only growing, they may be willing to lower their standards a bit (even subconsciously) if things do not work out with their “first picks” (or second picks).
In terms of how to reapply, you can simply state that you sent your resume on X date, you noticed that the job is still open, and you would like to bring to the firm’s attention a few changes you’ve made to your resume and re-express your interest in this position. Reapplying can increase your chances of landing interviews. You should not give up if you are not called for an interview the first time you apply.
If a firm does remember your background from prior review, you may be viewed as persistent and as truly interested in the position. Often times what makes a candidate very appealing in a firm’s eyes beyond just the academic and firm credentials they offer is the level of enthusiasm they bring to the new position. Firms want attorneys who genuinely want to come to work for them and who will be excited to join them. This sort of positive attitude toward an employer is something that employers find very attractive.
So, as a matter of practice, take a look at the positions for which you have applied, and for which you meet the qualifications, that are still listed. Reapply and increase your chances of being looked at on paper and in person.