You may put anything in first position of a main clause.
It becomes the topic of that main clause.
The only condition on that item in first position is that it must be one single item. So you can't put the subject and the accusative object there. Or an adverbial and the dative object or similar.
So let's assume die anzunehmen is indeed one single item. What is it then? It has an infinitive with zu, so it's an infinitive group. Let's use the expletive es in first position and reorder:
- Es würde dann wahrscheinlich jeder Kellner ablehnen, die anzunehmen.
So die anzunehmen is an infinitive group that tells what any waiter would deny to do.
You may still ponder about that die. What's that? Well, it's a pointing finger, a demonstrative pronoun. A those. Let's translate:
- Any waiter would likely deny to accept those then.
In lightly germanized English, the original sentence would read:
- To accept those, any waiter would likely deny then.