I've got everything working, though the method is still pretty sketchy. I've pieced together a script from bits of code here and there, mostly thanks to Tim Williams' comment. The program is indeed a version of Attachmate 2014, so that resource with its various guides was pretty helpful.
In short, my solution is this:
Private Sub CommandButton1_Click()
Dim app As Attachmate_Reflection_Objects_Framework.ApplicationObject
Dim frame As Attachmate_Reflection_Objects.frame
Dim terminal As Attachmate_Reflection_Objects_Emulation_IbmHosts.IbmTerminal
Dim view As Attachmate_Reflection_Objects.view
Dim screen As Attachmate_Reflection_Objects_Emulation_IbmHosts.IbmScreen
Dim cellData As String, w As String, x As String, y As String, z As String
Dim rCode As ReturnCode
Dim row As Integer
row = 3
Initialization stuff
Do
'Get the data from Excel
w = Worksheets("Calculator").Cells(row, 4).Value
x = Worksheets("Calculator").Cells(row, 5).Value
y = Worksheets("Calculator").Cells(row, 6).Value
z = Worksheets("Calculator").Cells(row, 7).Value
'Put the data into the appropriate fields
rCode = screen.PutText2(w, 22, 21)
rCode = screen.PutText2(x, 22, 47)
rCode = screen.PutText2(y, 22, 75)
rCode = screen.PutText2(z, 23, 20)
'Navigate and copy from the screens
rCode = screen.SendControlKey(ControlKeyCode_Transmit)
rCode = screen.WaitForText1(200, "4", 22, 4, TextComparisonOption_RegularExpression)
screen.SelectAll
screen.Copy
rCode = screen.WaitForHostSettle(30, 100)
rCode = screen.SendControlKeySync(ControlKeyCode_F3)
'Increment the counter to the next row
row = row + 1
'Paste into worksheet
With Worksheets("Data")
Range("A1").Select
.Paste
End With
Loop While Worksheets("Face").Range("D7").Value = "Searching..."
End Sub
And from there, excel uses basic IF statement to look at the pasted text and determine if the search term is in it. It goes through ~40 screens per minute which is comparable or slightly slower than a human but over a long period of time its definitely faster and easily more accurate.
My next step will be to rebuild it where it scrapes text fields and compares it to the search term itself instead of all the copy pasting nonsense that is the bulk of the time per loop.
EDIT: A few Weeks later...
I've made some optimizations using another very helpful vba function from the attachmate library, ".gettext(row, column, length)" and rather than copy & pasting, comparing the "gettext" (stored as a string) with the search term (also stored as a string from the excel worksheet). It hasn't had a noticeable impact on search speed, however. I've also added some logic to it that detects if it's on a blank screen (using .gettext), and skips to the next non-blank screen (the address of the next non-blank screen is shown on all blank screens so it's a simple matter of reading and interpreting that condition). All in all, it's doing the job and my colleagues (none of us are in IT or have knowledge of coding) are quite impressed. The next step will be scraping and cashing the screen info in a pseudo-database copy that I will build to mirror the live database which I can easily store in an excel sheet and use ctrl+f to search instead, and perhaps set it to update itself maybe once a week or once a month (since the database is rarely ever updated). All of this because we do not have a procedure to share database access and nobody cares enough to develop one.