No, you should not add RFC1918 networks to your SPF record. Instead, you should configure your Exchange server to omit SPF checks and allow relay for your web servers. If you trust your web applications, you could simply allow anonymous relay from your web servers.
Anonymous relay is a common requirement for many businesses that have
internal web servers, database servers, monitoring applications, or
other network devices that generate email messages, but are incapable
of actually sending those messages.
In Exchange Server, you can create a dedicated Receive connector in
the Front End Transport service on a Mailbox server that allows
anonymous relay from a specific list of internal network hosts.
The article describes this in more detail, but here's a summary of the Exchange Management Shell commands used for allowing anonymous relay from two servers (
New-ReceiveConnector -Name "Web Servers Relay" -TransportRole FrontendTransport `
-Custom -Bindings 0.0.0.0:25 -RemoteIpRanges 192.168.1.14,192.168.1.15
Set-ReceiveConnector "Web Servers Relay" -PermissionGroups AnonymousUsers
Get-ReceiveConnector "Web Servers Relay" `
| Add-ADPermission -User "NT AUTHORITY\ANONYMOUS LOGON" `
If you don't want to allow anonymous relay, your web applications would need to use authenticated SMTP, but as your original proposition was to add these servers to your SPF record, using anonymous relay is probably fine for you.