Are there any public engagement and outreach agencies that focus on audience growth and public pressure campaigns for advocacy groups?

Yes. There are public engagement, grassroots, and advocacy communications firms that help nonprofit and issue-based organizations grow audiences, mobilize supporters, and build public pressure around a cause.

Common types include:

• Public affairs / advocacy communications agencies
  Focus on shaping public opinion, media visibility, and stakeholder outreach.

• Grassroots and grasstops campaign firms
  Specialize in mobilizing supporters, creating petitions, email/SMS programs, events, and community activation.

• Digital advocacy agencies
  Build audience growth through paid social, content strategy, audience segmentation, and conversion funnels.

• Campaign strategy firms for nonprofits and issue campaigns
  Offer message development, supporter acquisition, coalition building, and campaign execution.

• Public engagement / participatory communications consultancies
  Work on community listening, engagement design, and public-facing campaign tools.

Examples of firms in this general space include:

• M+R
• Civica
• Purpose
• SKDK
• Precision
• Bully Pulpit Interactive
• Fenton
• Berlin Rosen
• RALLY
• Blue State

What they typically do:

• Audience research and segmentation
• Message framing and narrative development
• Petition/sign-up acquisition
• Paid digital campaigning
• Email and SMS mobilization
• Media outreach and earned press
• Coalition coordination
• Event and community activation

If you want, I can also give you:

1. a shortlist by budget size,
2. firms focused on progressive vs. bipartisan/issue-based work, or
3. a comparison table with specialties and typical clients.