Yes, there is a statutory time limit for the court reporter (either 90 or 120 days), but it is difficult if not impossible to enforce. If you check the Court of Appeal's on line docket for your case, it will show the date by which the Court expects the reporter to lodge the transcript. If you have concerns about progress, you can always call the court reporter and ask for a status report.
All of this assumes, of course, that you have paid the full required deposit for the transcript.
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I am presuming a couple of things here, first, that this is a civil matter and second, that the appeal is going to be heard by a court of appeal.
Did you deposit the reporters' transcript fees with your notice? Assuming you did, the court reporter has 30 calendar days from the date of receiving notice from the superior court clerk that the Notice of Designation of Record has been filed and the fees have been deposited for the preparation of the transcripts. The superior court cannot grant an extension to the court reporter. However, the court of appeal can.
If your case has been assigned a case number, you can check on what is happening in the court of appeal or supreme court by navigating this site: http://www.courts.ca.gov/