I figured this out on my own. The buttons in my datagrid call the following function, supplying it with the row specific parameter which is used on the server side to store the document in the proper place in the database.
It was necessary for me to view the source code to find the client id of the Uploader1_Insert control component and to set the uploader UploadType to "IFrame".
function DisplayUpload(RequestID) {
document.getElementById("<%=hdnRequestID.ClientID%>").value = RequestID;
document.getElementById("ctl00_ContentPlaceHolder1_Uploader1__Insert").click();
}