Model to detect specific semantic content without labeled data
First, this is not sentiment analysis, which is about quantifying &quot;affective states&quot;, i.e. something along the lines of &quot;like / don't like&quot;. This would rather correspond to a specific application of general text classification: predicting whether a sentence contains a contact request.
Now about the specific problem: you're right, it could be a good idea to start by identifying some clearly positive sentences using pattern matching. There are many approaches which can be used to expand this initial set of positive sentences:

semi-supervised learning
Using semantic similarity between words (for instance with word embeddings) to find more candidate terms for pattern matching
Using similarity between sentences to find potential new positive cases, then label these manually (this could be done iteratively)

Note that all the above approaches could work but they are biased in favor of the known positive cases, i.e. they risk missing positive cases which have little similarity with the original ones. The only option to avoid this issue is to take a random subset of sentences and label all of them as either positive or negative.