Basically, at that time, there was no reason for the FBI to issue a warrant. As the Federal Rules of Criminal Procedure states:
(d) Obtaining a Warrant.
(1) In General. After receiving an affidavit or other information, a magistrate judge—or if authorized by Rule 41(b), a judge of a state court of record—must issue the warrant if there is probable cause to search for and seize a person or property or to install and use a tracking device.
Using a private email server did not break any laws, as the article you cited states:
In Clinton’s defense, we should note that it was only after Clinton left the State Department, that the National Archives issued a recommendation that government employees should avoid conducting official business on personal emails (though they noted there might be extenuating circumstances such as an emergency that require it). Additionally, in 2014, President Barack Obama signed changes to the Federal Records Act that explicitly said federal officials can only use personal email addresses if they also copy or send the emails to their official account.
Because these rules weren’t in effect when Clinton was in office, "she was in compliance with the laws and regulations at the time," said Gary Bass, founder and former director of OMB Watch, a government accountability organization.
Eventually, the FBI managed to secure a search warrant to access emails between Hillary Clinton and her adviser Huma Abedin on a laptop belonging to Anthony Weiner.
The new documents reveal FBI agents secured the warrant after assuring a federal judge there was “probable cause” to believe the emails, contained on a laptop belonging to Anthony Weiner, a former US congressman and estranged husband of Ms Abedin, contained illegal and classified materials.
Quoting from this Politico article, court filings unsealed showed that the FBI believed "that the laptop was likely to contain evidence of illegal possession of classified information":
"There is probable cause to believe that the Subject Laptop contains evidence, contraband, fruits, and/or other items illegally possessed in violation of 18 U.S.C. § 793 (e) and (f)," an FBI agent wrote, citing felony Espionage Act provisions for illegal possessions of classified information.