Start Application dependencies before main Applciation is started

Where does :my_dependency come from. Unless it’s an elixir core or otp application it should generally be listed in your mix.exs deps. :extra_applications is for those elixir core/otp applications you might depend on.

How will this work if MyDependency starts a named process? The process will start when the dependency starts and then you’ll try to start it again in your supervision tree.

It is an umbrella library. Yes, it does have a GenServer but it is not an OTP application.
I have it in my mix deps function as well:

  defp deps do
    [
      # Phoenix
      {:phoenix, "~> 1.7.2"},
      {:phoenix_html, "~> 3.3"},
      {:phoenix_live_reload, "~> 1.2", only: :dev},
      {:phoenix_live_view, "~> 0.18.16"},
      {:floki, "~> 0.34"},
      {:esbuild, "~> 0.7", runtime: Mix.env() == :dev},
      {:tailwind, "~> 0.2.0", runtime: Mix.env() == :dev},
      {:telemetry_metrics, "~> 0.6"},
      {:telemetry_poller, "~> 1.0"},
      {:jason, "~> 1.2"},
      {:plug_cowboy, "~> 2.5"},

      # Umbrella deps
      {:manager, in_umbrella: true}
    ]
  end

Yes, this is the case.

I am guessing the issue here is that the process is not yet started (judging from the error)

Shouldn’t it be:

Supervisor.start_link(children, opts)   
{:ok, things} = MyDependency.list_things()

The start function/callback must return a link to a Supervisor. That won’t be the case if I follow your suggestion.

with {:ok, pid} <- Supervisor.start_link(children, opts) do
  {:ok, things} = MyDependency.list_things()
  {:ok, pid}
end

You need to start the supervisor (and therefore its children) before you can expect the children to run.