Internet Explorer:
You can click using the id
ie.document.querySelector("#export").click
Or attribute = value selector
ie.document.querySelector("[onclick='core.essres.exportres();']").click
Or even try executing the onclick function direct
ie.document.parentWindow.execScript "core.essres.exportres();"
The other answer shows you how to handle the SaveAs dialog.
Direct download:
You could also use dev tools when clicking download to see if there is an url in the network tab associated with the download which you could pass direct to urlmon or binary download
Selenium:
You could switch to selenium vba and have a choice of browsers such as chrome, where you don't have the save/open dialog, where you can specify a default download location, or even just leave to download to your current default
Option Explicit
'download selenium https://github.com/florentbr/SeleniumBasic/releases/tag/v2.0.9.0
'Ensure latest applicable driver e.g. ChromeDriver.exe in Selenium folder
'VBE > Tools > References > Add reference to selenium type library
Public Sub DownloadFile()
Dim d As WebDriver
Set d = New ChromeDriver
Const URL = "url"
With d
.Start "Chrome"
.get URL
.FindElementById("export").Click
Application.Wait Now + TimeSerial(0, 1, 0) ' leave time to download before exiting or _
loop download folder checking for when new file appears (or expected file by name/part of file name
.Quit
End With
End Sub