I took Dave's suggestion and modified ConnectionString() and Open() with a few minor alterations.  Also added the enum and a couple of member vars.  VB.NET 1.1 code posted below. 

Code:

    Private m_authType As AuthenticationType
    Private m_strSessionId As String
    Private m_strUrl As String

    Private Enum AuthenticationType
        SessionId
        Username
    End Enum

    Public Property ConnectionString() As String Implements System.Data.IDbConnection.ConnectionString
        Get
            Return m_strConnection
        End Get
        Set(ByVal Value As String)
            'parse
            m_strConnection = Value
            m_htbConnParms = ParseConnectionString(Value)

            ' I'm resetting the sessionid parameter because I found the '=' symbol at the end of the SessionId got parsed out in ParseConnectionString().
            ' One could modify that accordingly to prevent this; I was lazy and hardcoded it here for test purposes.
            m_htbConnParms("sessionid") = "qrKAqvXXOOvnVBptJ0UdrmQVLCitqQdkifCrjoL_2cDbTxOkOEHQ.bSobBGwX66rDxpsMAr0i9BTm_rlx9Z86WpYnKO9AsNCSeX5jsUoLXQ="

            ' Username and Password-Style Authentication (note keys are lower case)
            If m_htbConnParms.ContainsKey("user id") And m_htbConnParms.ContainsKey("password") Then
                m_strUsername = m_htbConnParms("user id").ToString
                m_strPassword = m_htbConnParms("password").ToString
                m_authType = AuthenticationType.Username
            End If

            ' SessionId and Server URL Authentication (note keys are lower case)
            If m_htbConnParms.ContainsKey("sessionid") And m_htbConnParms.ContainsKey("serverurl") Then
                m_strSessionId = m_htbConnParms("sessionid").ToString
                m_strUrl = m_htbConnParms("serverurl").ToString
                m_authType = AuthenticationType.SessionId
            End If

            If m_htbConnParms.ContainsKey("querybatchsize") Then
                m_intQueryBatchSize = Convert.ToInt32(m_htbConnParms("querybatchsize"))
            End If

            If m_htbConnParms.ContainsKey("retrievechunksize") Then
                m_intRetrieveChunkSize = Convert.ToInt32(m_htbConnParms("retrievechunksize"))
            End If

            If m_htbConnParms.ContainsKey("sendcompressedrequest") Then
                m_svcSforce.SendCompressedRequest = Boolean.Parse(m_htbConnParms("sendcompressedrequest").ToString)
            End If

            If m_htbConnParms.ContainsKey("acceptcompressedresponse") Then
                m_svcSforce.AcceptCompressedResponse = Boolean.Parse(m_htbConnParms("acceptcompressedresponse").ToString)
            End If

            If m_htbConnParms.ContainsKey("commandtimeout") Then
                Me.CommandTimeout = Convert.ToInt32(m_htbConnParms("commandtimeout"))
            End If
        End Set
    End Property

    Public Sub Open() Implements System.Data.IDbConnection.Open
        Try
            With m_svcSforce
                If m_enmConnState = ConnectionState.Closed Then
                    With m_svcSforce
                        m_enmConnState = ConnectionState.Connecting

                        ' Instantiate SessionHeader
                        .SessionHeaderValue = New Sforce.SessionHeader

                        ' Authenticate via Username/Pwd or SessionId/Url
                        If m_authType = AuthenticationType.Username Then
                            Dim objLR As Sforce.LoginResult = .login(m_strUsername, m_strPassword)

                            .SessionHeaderValue.sessionId = objLR.sessionId
                            .Url = objLR.serverUrl

                            m_strUserId = objLR.userId
                        ElseIf m_authType = AuthenticationType.SessionId Then
                            .SessionHeaderValue.sessionId = m_strSessionId
                            .Url = m_strUrl

                            m_strUserId = m_svcSforce.getUserInfo().userId
                        End If

                        ' Query Options
                        If m_intQueryBatchSize > 0 Then
                            .QueryOptionsValue = New Sforce.QueryOptions
                            .QueryOptionsValue.batchSize = m_intQueryBatchSize
                            .QueryOptionsValue.batchSizeSpecified = True
                        End If

                        ' Set ConnectionState to Open
                        m_enmConnState = ConnectionState.Open
                    End With
                Else
                    Throw New InvalidOperationException("Connection already open.  State: " & m_enmConnState.ToString)
                End If
            End With
        Catch ex As Exception
            m_enmConnState = ConnectionState.Closed
            Throw ex
        End Try
    End Sub

 
Along the way, however, I found that one could easily enable the existing SforceConnection object (i.e., no modifications necessary to ADO.NET provider) to assume a new SessionHeader.  This is done by first logging in "traditionally" with a username and password (ideally, an API designated account), then resetting the SforceConnection's SessionHeader value to one that uses the SessionId passed in via weblink.  Note that I didn't need to change the Url property, but that could be accomplished by passing the API Url via weblink, as well. 

This approach is posted below in a C# 2.0 sample method below called UpdateAccount().

Code:

using Salesforce.SforceClient;
using Salesforce.SforceClient.Sforce;

    private void UpdateAccount(string accountId, string newValue)
    {
        // Open SFDC Connection using an API Account
        SforceConnection conn = new SforceConnection("User ID=api@yourorg.com;Password=xxxxxxx");
        conn.Open();

        // Set the SessionHeader to the session of the user that clicked the weblink
        conn.SforceService.SessionHeaderValue = new SessionHeader();
        conn.SforceService.SessionHeaderValue.sessionId = Request.QueryString["SFDCSessionId"];

        // This update will occur in the authentication context of the user that clicked the weblink
        SforceCommand cmd = new SforceCommand("UPDATE Account SET SomeField__c = @SomeField__c WHERE Id = @Id", conn);
        cmd.Parameters["@Id"].Value = accountId;
        cmd.Parameters["@SomeField__c"].Value = newValue;

        int result = cmd.ExecuteNonQuery();
    }