Father was a veteran. I paid for the plot, opening & closing. Could I have gotten that for free? How can I get his headstone?

Usually the funeral home will apply for the plaque for you at the time of the funeral. You have to fill out the form mentioned. The VA will send the plaque to the cemetery or funeral home at the address given on the form. There are (at least) two choices. A bronze metal one that needs to be placed on a stone ( at your expense) or a marble or granite one that can go directly on the grave. Usually whichever office who receives the plaque will contact the name on the form to let them know the plaque has arrived.
The VA will have a record of who they sent it to and when.
Here is a link that gives more information, phone numbers and a link to the form.
Remember that the cemetery where your dad is buried may have rules on what is placed in the cemetery. (Flat vs upright, size of stone under plaque). If you walk around the cemetery you will find examples.

https://www.cem.va.gov/hmm/

Is he in a Veterans cemetary? If so is it a County one, I was under the impression county plots are free but you pay for opening and closing it. You also get a plaque free. National cemetery everything is free you just need to be cremated.

You need to contact your County VA office and tell them Dad never received his plaque. My in-laws are buried in a National Cemetery. There was a representative of the cemetery there that took all the info for the plaque on the day of burial. The plaque was installed a few weeks later.

There are cemeteries the VA has and a veteran can be buried there. If he was buried in another cemetery you can get a plaque for the headstone but the VA will not pay for a headstone.
My Husband was a veteran, he had a plot in a cemetery I paid for the opening and closing, I paid for the vault and the casket. The only thing that the VA provided was the plaque that is on the stone that marks his grave.
Had I chosen to have him buried in the VA cemetery I could have but it is 2 hours away and he/we already owned the plot.
There is a form you can fill out for the plaque contact the VA or maybe the local VFW would help you out

What type of cemetery is your dad buried in? If he is in a private cemetery then no, you couldn’t haven’t gotten him buried for there free. The VA does not pay for cemetery plots & the opening and closing of a plot in private cemeteries. There is a burial allowance but amounts have changed in recent years. The current allowance wouldn’t even pay for half the cost of a cemetery plot. And I do believe you only had 2 years from the date of his burial to file a reimbursement claim. Unless this was a service-connected death.

Correct guestion