Hi @ch38,
Have you created the button with column/list formatting?
If that is the case you can use the '@me special string' value in a custom parameter called emailaddress. When that custom parameter is added you can retrieve it via the below expression:
triggerOutputs()['queries']['customparameter']
Below is an example
1. The outputs would look like the below.

2. Below is an example of json which can add such a button with emailaddress & currentweb query parameters.
{
"$schema": "https://developer.microsoft.com/json-schemas/sp/v2/column-formatting.schema.json",
"elmType": "button",
"style": {
"border-radius": "5px",
"margin": "5px 0px",
"padding": "0px",
"visibility": "visible"
},
"attributes": {
"class": "ms-bgColor-themePrimary"
},
"children": [
{
"elmType": "a",
"txtContent": "E-Mail Me",
"style": {
"text-decoration": "none",
"padding": "10px 0px",
"width": "100%"
},
"attributes": {
"href": "='https://WhenarequestisreceivedGETurl&emailaddress=' + @me + '¤tWeb=' + @currentWeb",
"target": "_blank",
"class": "ms-fontColor-white"
}
}
]
}

More detailed steps can be found in this blog I have written about this same approach:
https://www.expiscornovus.com/2023/01/20/e-mail-me-button-on-page/