Learn French: J'appelais pour solliciter une recommandation. - I was calling to request a recommendation.

This is the "imparfait d'atténuation" (imperfect of politeness). Using the imperfect instead of the present (j'appelle) creates a little psychological distance, making the request sound softer and less demanding. It maps perfectly to how we say "I was calling..." in English when making a polite request.

Could I just say J'appelle pour... instead?

Yes, J'appelle pour... is grammatically correct and very common. However, it is more direct. If you are asking for a favor like a recommendation, the imperfect J'appelais adds a layer of tact that the present tense lacks.

Is solliciter just a formal way of saying demander?

Exactly. Solliciter is a formal verb meaning "to request" or "to seek". While you would use demander in everyday situations, solliciter pairs beautifully with the professional, polite tone established by the imperfect J'appelais.

Why is there only one "l" in J'appelais but two in J'appelle?

For stem-changing verbs like appeler, the consonant is only doubled when the syllable that follows contains a silent "e". Because the imperfect ending -ais has a pronounced vowel sound, the "l" remains single.