You can eliminate the need for the local DSN on each machine by using a bit of code in the Access database. You do of course still need the appropriate ODBC driver installed but I expect the one you need is probably standard on Windows anyway.
A local DSN will be required to initially create the database.
The following in what I use for dynamically relinking to a MySQL database, so you will need to edit it accordingly. The code is called from the AutoExec macro or can be run manually or from a form.
Note that this is not my code but I've been using it for a very long time and don't recall where I originally got it. All I've done is edit it to suit my requirements.
Option Compare Database
Public Function ReLinkTables()
Dim dbPUBS As DAO.Database
Dim tdfPUBS As DAO.TableDef
Dim strTable As String
Dim strConnect As String
Dim InFile As Integer
' Set the following variables tosuit your DB connection
Dim Server As String
Dim Database As String
Dim User As String
Dim Password As String
On Error GoTo 0
Set dbPUBS = Nothing
Set dbPUBS = CurrentDb
strConnect = "DRIVER={MySQL ODBC 3.51 Driver};" _
& "SERVER=" & Server & ";" _
& "DATABASE=" & Database & ";" _
& "UID=" & User & ";" _
& "PWD=" & Password & ";" _
& "OPTION=" & 1 + 2 + 8 + 32 + 2048 + 16384
' Refresh Access linked tables
For Each tdfPUBS In dbPUBS.tabledefs
' Only attempt to refresh link on tables that already
' have a connect string (linked tables only)
If Len(tdfPUBS.Connect) > 0 Then
strTable = tdfPUBS.Name
' Set the tables connection string
tdfPUBS.Connect = strConnect
' and refresh the link
tdfPUBS.RefreshLink
End If
Next
' Refresh Connect String for all Pass-Through Queries
'strMsg = "Refreshing links for all Pass Through Queries."
'DoCmd.Echo True, strMsg
'For Each qdfPUBS In dbPUBS.QueryDefs
'If Len(tdfPUBS.Connect) > 0 Then
'qdfPUBS.Connect = strConnect
'End If
'Next
End Function