No, you have it wrong... you need to file the pre-requisite FIRST (with the subpoena attached). This gets served on the other lawyers in the case. If THEY do not object after 20 days, THEN you can serve the subpoena on the party whose records you want. See the full Rule below.
Rule 4009.22. Service of Subpoena.
(a) The party seeking production may serve on the person named in the subpoena a copy of the subpoena only if it is identical to the subpoena attached to the notice of intent to serve the subpoena and if the party seeking production has filed of record a certificate that
(1) the notice of intent to serve a subpoena was mailed or delivered to each party at least twenty days prior to the date on which the subpoena is sought to be served,
(2) a copy of the notice of intent, including the proposed subpoena attached to the notice of intent, is attached to the certificate,
(3) no objection to the subpoena has been received, and
(4) the subpoena which will be served is identical to the subpoena which is attached to the notice of intent to serve the subpoena.
For the form of the certificate, see Rule 4009.25.
The twenty-day advance notice is for the benefit of the parties and not the person served. The twenty-day notice period may be waived and the certificate modified accordingly.
(b) The subpoena shall be issued as provided by Rule 234.2(a) and shall be served in the manner provided by Rule 234.2(b).