اليوم أنتظر جوابا للمشكلة التي لدينا في المكتب.

Yes, word order is flexible. اليوم is a time adverb (ظرف زمان) and is often placed at the beginning for emphasis or to set the scene: “Today…”. You could also say:

• أنتظر اليوم جوابًا... (more neutral)
• اليومَ أنتظر جوابًا... (with the case ending, more formal/fully-voweled)

In Arabic the verb form usually includes the subject. أنتظرُ is Form VIII (اِفْتَعَلَ) in the present tense, and the prefix أ- marks 1st person singular (“I”).
So (أنا) أنتظرُ is correct, but أنا is often omitted unless you want emphasis/contrast.

The Arabic present tense (المضارع) can cover both:

• habitual/general: “I (usually) wait”
• ongoing/current: “I am waiting” Context decides. With اليوم it often sounds like “Today I’m waiting (for…)” / “Today I’m expecting…”. It can also imply near future expectation, similar to “I’m waiting for a reply (that should come).”

جوابًا is the direct object (مفعول به) of أنتظر (“I’m waiting for an answer”), so it is accusative (منصوب). In fully-voweled Arabic it would be:

• أنتظرُ جوابًا

The -an sound (ًا) is the case ending for an indefinite accusative noun.

Most everyday Arabic writing omits short vowels and case endings. So you’ll often see جوابا instead of جوابًا. The grammar is still “there” even if the marks aren’t written.

Often both can translate as “answer,” but there’s a tendency:

• جواب: a reply/response (often to a message, request, question)
• إجابة: an answer in the sense of “the solution/answering” (often to a question/test, or the act of answering)

In this sentence جواب sounds very natural: “a reply/response regarding the problem.”

لِـ here means “to / regarding / about.” So جوابًا للمشكلة is literally “an answer to/regarding the problem.”
The لِـ attaches to the noun, and with الـ it becomes لِ + الـ = لِلـ:

• لِلمشكلة (li-l-mushkilah)

They don’t have to match. The sentence means “an answer (some answer) to the problem (a specific known problem).”
If you wanted “the answer,” you could say:

• اليوم أنتظرُ الجوابَ للمشكلة... But Arabic often prefers a different structure for “the answer to the problem,” such as an iḍāfa-like phrasing:
• أنتظرُ جوابَ المشكلةِ (common and compact)

التي is the relative pronoun for feminine singular antecedents. Since مشكلة is feminine singular, you use التي (not الذي, which is masculine singular).
The phrase التي لدينا في المكتب is a relative clause describing المشكلة:

• “the problem that we have in the office”

لدينا means “we have” / “we possess” / “there is with us.” Grammatically it’s based on لدى (“with/at/in the possession of”) + the pronoun نا (“us/our”).
So لدينا مشكلة = “We have a problem.”

It modifies the whole idea of “the problem we have,” meaning the problem exists in the office context:

• “the problem that we have at the office”

It does not mean you “have an office.” If you wanted “our office,” you’d usually say:

• في مكتبِنا (“in our office”)

A simple breakdown:

• اليوم: time adverb (ظرف زمان) = “today”
• أنتظرُ: present verb, 1st person singular = “I wait / I’m waiting”
• جوابًا: direct object (مفعول به منصوب) = “an answer”
• لِلمشكلةِ: prepositional phrase (جار ومجرور) = “to/regarding the problem”
• التي لدينا في المكتب: relative clause (جملة صلة الموصول) describing المشكلة:
  • التي: relative pronoun (fem. sing.)
  • لدينا: “we have”
  • في المكتب: “in/at the office”