A Website I use received subpoenas from the US Department of Justice for data related to certain specified usernames. What standard of evidence must that subpoena have met and how does it compare to the probable cause required for a search warrant?
Probable cause is not required to issue a subpoena.
But, a recipient of a subpoena (or a party to a case in which a subpoena is issued, or a person whose records a subpoenaed), may file a motion to quash the subpoena in the court that issued it, before the information is disclosed pursuant to the subpoena.
A claim that the subpoenaed material is privileged is always a legitimate reason to quash a subpoena. A subpoena can also be quashed on the grounds that the materials sought are unrelated to the matter for which it is issued, or that it is unduly burdensome in a manner unrelated to a legitimate interest in obtaining the information sought (this list of objections is not exhaustive).
You can see an example of a motion to quash a subpoena duces tecum from a federal grand jury, together with the subpoena itself attached to the motion at the end of the linked file as an exhibit, at this link. The federal court rule governing subpoenas in federal criminal cases is Federal Rule of Criminal Procedure 17. In federal court civil cases, Federal Rule of Civil Procedure 45 applies.
Short of actually quashing the subpoena, the court issuing it could also modify the obligation by imposing a protective order on the materials to be disclosed (e.g. trade secrets or private financial information). This would prevent the materials disclosed from being shared with people other than the recipient.
Incidentally, the penalty for failing to respond to a subpoena is for the person to whom it is directed to hold the person in contempt of court, punishable by incarceration or a fine (often a fixed amount per day). Often, if testimony is part of what is required, a warrant for the arrest of the person who fails to appear is summarily issued by the court.
Footnote regarding terminology
Strictly speaking, the word "subpoena" unmodified, refers to a subpoena directing the person to whom it is addressed to testify under oath at a certain place and time, sometimes a grand jury, sometimes a trial in a court, sometimes a deposition, sometimes an arbitration, and sometimes an interview with a government official.
When a subpoena is asking for the turnover of records or things or data it is strictly speaking called a "subpoena duces tecum" (from the Latin duces tecum, meaning "you shall bring with you").
Historically, in response to the subpoena duces tecum, the custodian of the records would appear personally at a deposition or grand jury proceeding or trial with the records or things and would testify under oath briefly at that time to authenticate them.
These days, the federal rules of evidence and most state rules of evidence provide that the custodian of records can sign an affidavit, or a declaration (i.e. a statement made under penalty of perjury which is not notarized), regarding the authenticity of the records (and their status as "business records" or "public records" where appropriate). Then, this affidavit or declaration is delivered along with the records by a courier or mail, in lieu of actually providing live testimony regarding the authenticity and business records status of the records.
Delivery of documents by courier or mail or electronically, accompanied by an affidavit or declaration is what is usually done these days, except in exceptional cases where the authenticity and/or business records status of the records is genuinely in dispute and complicated in some way. For example, in person testimony of the custodian of records might be required in a case where there are allegations of forgery of documents or that digital files were hacked.