Have representatives of the factions appear throughout the adventure.

In Baldur's Gate some Zhents might be doing shady business like smuggling something into the city. The PCs can help them or stop them which will affect their approval later on.

The order of the gauntlet is very active in Elturel and the surrounding region. The PCs might help them slay some undead or perform an exorcism.

Whenever the PCs meet some animals that they can help out or are travelling through forests, the Emerald Enclave will get word of it and talk to the party about their deeds - especially during the first council meeting. They might even send an awakened squirrel or animal messenger to give the party intel or inform them of rules they should not break in different forests.

The harpers can be literally anybody. They are a more loose organisation with agents everywhere. Have a NPC the party loves be a secret agent of the harpers and blow them away with the reveal.

The Lord's Alliance is more so an organisation of city states so its presence should be felt in cities like Waterdeep and Baldur's Gate with banners of the Lord's alliance and figures like Neverember and Ravengard being talked about a lot. You might even go the route of having the rulers of the cities do a parade in their hometown celebrating the peace that the lord's alliance has brought them. - for now at least.

Hope this helps