Learn Hardcore Korean: hoesa-eseo piryohan jangbireul sasseoyo.

-에 marks a static location (“at/to”).
-에서 marks where an action takes place.
Here you’re talking about the place where you bought something, so you need 회사에서 (“at the company”) rather than 회사에.

What does 필요한 mean, and why is it placed before 장비?

필요하다 means “to be necessary.” To turn it into a modifier for a noun (an attributive form), you:

Drop -다 → 필요하
Attach -ㄴ (since the stem ends in a vowel) → 필요한
So 필요한 장비 literally means “equipment that is necessary.”

Why is 장비 followed by -를 (as in 장비를)?

-를 is the object particle used after a noun ending in a vowel. It marks 장비 as the direct object of 샀어요 (“bought”). If the noun ended in a consonant, you would use -을 instead.

What is the nuance or politeness level of 샀어요?

샀어요 is the polite (해요) form in past tense of 사다 (“to buy”).

It’s appropriate for general situations with strangers or acquaintances.
More formal: 샀습니다
Less formal/familiar: 샀어

Where is the subject in 회사에서 필요한 장비를 샀어요?

Korean often omits pronouns when they are clear from context. Here, the subject “I” (저는/내가) is implied. Fully spelled out it would be:
저는 회사에서 필요한 장비를 샀어요.

Could we say 회사에 필요한 장비를 샀어요 instead? What’s the difference?

회사에 필요한 장비 means “equipment necessary for the company” (company as beneficiary).
회사에서 필요한 장비 means “equipment needed at the company” (location of necessity).
Depending on nuance you choose -에 or -에서.

Is it OK to rearrange the sentence to 필요한 장비를 회사에서 샀어요?

Yes. Korean word order is fairly flexible. Moving 필요한 장비를 to the front only shifts emphasis onto the equipment. The meaning remains “I bought the necessary equipment at the company.”

What part of speech is 필요하다 in Korean?

필요하다 is a descriptive verb (often called an adjective in English). It describes a state (“to be necessary”). Descriptive verbs:

Conjugate like verbs (attach tense endings)
Modify nouns by changing -다 to -ㄴ/은/는 forms (e.g. 필요한, 예쁜, 좋은)