Does Forecaster activate vision when played alone, and if you have multiple Forecasters in play, does playing a mech trigger vision multiple times?

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Yes. When you play Forecaster it will get Vision and that Vision will trigger as it enters, letting you look at the top card and optionally recycle it. If you control multiple permanents that grant Vision (for example multiple Forecasters), each instance of Vision on the entering Mech triggers separately, and each trigger resolves independently.

Playing Forecaster alone: its printed static effect "Your Mechs have [Vision]" applies to itself (it's a Mech), so its Vision trigger is created when it enters the Board and you may look at the top card and may recycle it. Multiple sources: if multiple permanents grant Vision to a Mech, each instance of Vision creates a separate trigger when that Mech is played; you may resolve them separately and choose to recycle independently for each trigger. If you decline to recycle on a trigger and nothing else happens between triggers resolving, each Vision trigger will see the same top card [817.2.b].

You play Forecaster. As it enters the Board, it becomes a permanent and its characteristics include its printed static ability that says "Your Mechs have [Vision]" [817.3]. Because Forecaster is a Mech, the static effect grants Vision to that permanent as it is on the Board; Vision is a triggered ability whose trigger is the permanent entering the Board [817.1.c] [817.1.b]. A Vision trigger for that permanent is created and placed on the chain. When that trigger resolves, you look at the top card of your Main Deck and may recycle it [817.1.b]. If you control multiple permanents that grant Vision to Mechs, each instance of Vision on the entering Mech creates its own trigger; those triggers resolve separately and you may choose to recycle or not for each one [817.2.a]. If you decline to recycle on an earlier Vision trigger and nothing else happens between triggers resolving, later Vision triggers will still see the same top card [817.2.b].

If a Mech that would get Vision is not on the Board when its Vision triggers resolve (for example it was moved to a non-Board Zone and returned), the triggers created when it entered still resolve; but since Vision triggers are tied to the entering event, typical timing interactions (like zone changes making objects different objects) could affect other interactions—however the core point remains that each instance of Vision creates a separate trigger on entry [817.1.c]. If an effect removes or adds Vision to a permanent between the moment it entered and trigger finalization, the presence/absence of Vision on the permanent at trigger creation matters because Vision is a characteristic of the permanent and is what makes the trigger exist in the first place [817.3]. If you control multiple Vision-granting permanents and choose not to recycle on one trigger, you can still choose to recycle when a subsequent Vision trigger resolves (choices are independent) [817.2.a]. If nothing else changes between multiple Vision triggers resolving, they will all look at the same top card if you declined to recycle earlier triggers [817.2.b].

Vision is a triggered ability keyword that is present on permanents and is functionally short for "When this is played, look at the top card of your Main Deck. You may recycle it." The trigger event for Vision is the permanent entering the Board [817.1.b] [817.1.c]. The rules also state that Vision and whether a permanent has Vision are characteristics of that permanent and may be checked by other Game Effects [817.3]. Forecaster's text says "Your Mechs have [Vision]." That is a continuous/static effect printed on the permanent that gives the Vision characteristic to Mechs you control. When you play Forecaster, it enters the Board as a Mech and, because the permanent's characteristics include the printed static effect that grants Vision to Mechs, the entering Forecaster will have Vision and thus create a Vision trigger on entry [817.3] [817.1.c] [817.1.b].

If you control multiple permanents that grant Vision to Mechs (for example multiple copies of Forecaster), each granted instance of Vision on the entering Mech creates its own trigger — the rules explicitly state multiple instances of Vision trigger separately and that the player may choose to recycle or not for each instance independently [817.2.a]. If you decline to recycle on an earlier Vision trigger and nothing else happens between triggers resolving, each instance of Vision will still see the same top card when they resolve [817.2.b].

Therefore: playing Forecaster alone will produce a Vision trigger for that Forecaster as it enters; playing a Mech while you control multiple Vision-granters will produce multiple separate Vision triggers, each resolved independently.