It depends upon the jurisdiction, however, generally, once a Will is filed in the Probate or Surrogates court it is a matter of public record. I don't know if any or all of them are available on line. I doubt it.
When a person dies with a will, the will has to be "admitted to Probate" in order to be effective. I don't know about your state but, in CT, all wills that have been admitted to Probate are public record, available to anyone. Someone can go to the Court and ask to see a will or can request a copy by mail for a fee. To my knowledge, there is no record kept of who looks at documents. You should check with your local or county Probate Court.
In WA, once a will has been filed in court for probate, the will is a public record and can be reviewed by anyone. In WA, wills are probated in the county's superior court.
Some WA's courts do have court records online. However, there is a fee to view the document online. Because there is payment, there will be a record of who paid to see and print the documents via the internet.
Viewing the documents at the court's clerk office is free. There is a charge for printing the documents. I have never had to enter any ID when viewing court records at the public computers at court clerk's office. When I print out court documents at the clerk's office, there usually is a form asking for my name. That is just so that I can use my name to pick up the documents (instead of picking up someone else's documents). A person wanting to remain unknown can make up a name.
I do not know the inner workings of the courts' computer systerms. I imagine that the systems, like any other computer systems, have the capability of tracking from where the requests for specific records originated.
WA has a program that allows people to give a will to the court clerk for safekeeping. However, no one, but the person depositing the will, has access to read those documents unless the testator is dead and there is a probate proceeding. There is a charge each time the will is retrieved or deposited.
Because wills being probated are public records, persons wanting privacy in their probate proceedings often use other methods to dispose of the assets.