Removing Items in a List While Iterating Through It with For Each Loop

Go for safe and make a copy with ToList():

For Each Needed In NeededList.ToList()
    Dim Ticker = Needed.Split("-")(0).Trim()
    ...
    If dr.HasRows Then
        NeededList.Remove(Needed)
    End If
Next

You can use a For loop iterating through every index with Step -1.

For i as Integer = NeededList.Count - 1 to 0 Step -1

    Dim Needed = NeededList(i)

    'this is a copy of your code
    Dim Ticker = Needed.Split("-")(0).Trim()
    Dim Year = Needed.Split("-")(1).Trim()
    Dim Period = Needed.Split("-")(2).Trim()
    Dim Table = Needed.Split("-")(3).Trim()

    Dim dr As OleDbDataReader
    Dim cmd2 As New OleDb.OleDbCommand("SELECT * FROM " & Table & " WHERE Ticker = ? AND [Year] = ? AND Period = ?", con)
    cmd2.Parameters.AddWithValue("?", Ticker)
    cmd2.Parameters.AddWithValue("?", Year)
    cmd2.Parameters.AddWithValue("?", Period)
    dr = cmd2.ExecuteReader

    'MODIFIED CODE
    If dr.HasRows Then NeededList.RemoveAt(i)

Next i

No you can't do that using a for each, but you can do that using the old fashioned for .. loop.
The trick is to start from the end and looping backwards.

For x = NeededList.Count - 1 to 0 Step -1
    ' Get the element to evaluate....
    Dim Needed = NeededList(x)
    .....
    If dr.HasRows Then
        NeededList.RemoveAt(x)
    End If
Next

You need to approach the loop in this way because you don't risk to skip elements because the current one has been deleted.

For example, suppose that you remove the fourth element in the collection, after that, the fifth element becomes the fourth. But then the indexer goes up to 5. In this way, the previous fifth element (now in fourth position) is never evaluated. Of course you could try to change the value of the indexer but this ends always in bad code and bugs waiting to happen.