How can I request some time from the judge, in order to find a new attorney?

If your attorney just withdrew, you will be given 21 days to obtain a new attorney or file your own supplementary appearance. 

"Unless another attorney is, at the time of such withdrawal, substituted for the one withdrawing, the party shall file in the case within 21 days after entry of the order of withdrawal a supplementary appearance, stating therein an address at which the service of notices or other documents may be had upon him or her" Ill. Sup. Ct. R. 13

If the 21 days have already passed you can simply ask for more time which courts will grant "for good cause shown."

“The court, for good cause shown on motion after notice to the opposite party, may extend the time for filing any pleading or the doing of any act which is required by the rules to be done within a limited period, either before or after the expiration of the time.” Ill. Sup. Ct. R. 183 (eff. Feb. 16, 2011).

“On good cause shown, in the discretion of the court and on just terms, additional time may be granted for the doing of any act or the taking of any step or proceeding prior to judgment.” 735 ILCS 5/2-1007

Be aware that the court is not obligated to grant your continuance so be on your best behavior. 

“A litigant has no absolute right to a continuance; rather, the decision to grant or deny a continuance is within the sound discretion of the trial court.” In re Marriage of Ward, 282 Ill. App. 3d 423, 430 (1996)

Good luck!

You should absolutely show up at your next court date and advise the Judge that you are in the process of hiring an attorney. He/She will give you 28 days to find an attorney.