Летом у нас будет практика в школе, и во время практики моя подруга будет работать с детьми и писать отчёты.

Летом is the instrumental form of лето, and Russian often uses the instrumental case to mean in/during a season:

• зимой = in winter
• весной = in spring
• летом = in summer
• осенью = in autumn

So Летом means In the summer / During the summer.
You generally do not say в лето for this meaning.

Here у нас means something like for us, we have, or in our case.

In this sentence, у нас будет практика literally means with us there will be practice, but natural English is more like:

• we’ll have a practicum/internship
• we have a school placement coming up

Russian often uses у + genitive to express possession or something connected with a person/group:

• у меня есть книга = I have a book
• у нас будет экзамен = we will have an exam
• у нас будет практика = we will have a practicum/internship

Because практика here is a noun, not a verb. It means practical training, internship, placement, or practicum.

So:

• у нас будет практика = we’ll have a practicum / practical training
• not necessarily we will be practicing

Russian often uses a noun where English might use a more verbal expression.

In this context, практика usually means a period of required practical work as part of study or training.

Depending on context, it can be translated as:

• practicum
• internship
• teaching practice
• work placement
• practical training

Since the sentence says в школе, it probably means some kind of school placement or teaching practicum.

Because в meaning in is followed here by the prepositional case when talking about location.

• школа = school
• в школе = in the school / at school

So практика в школе means practicum at a school or in a school.

Russian uses во instead of в before some consonant clusters to make pronunciation easier.

So the fixed expression is:

• во время = during

You should learn во время as a set phrase.

Examples:

• во время урока = during the lesson
• во время практики = during the practicum

Because во время requires the genitive case.

• практика = nominative
• практики = genitive

So:

• во время практики = during the practicum

This is a very common pattern:

• во время занятия = during the class
• во время каникул = during the holidays
• во время работы = during work

This is the compound future of an imperfective verb:

• работать → будет работать
• писать → будет писать

Russian uses this form when the action is viewed as:

• ongoing
• repeated
• habitual
• not focused on completion

Here that makes sense: during the practicum, she will be working with children and writing reports as regular activities.

If Russian wanted to focus more on completed results, it might use perfective verbs instead.

Because the preposition с meaning with normally takes the instrumental case.

• дети = children
• с детьми = with children

So:

• работать с детьми = to work with children

More examples:

• с другом = with a friend
• с учителем = with the teacher
• с родителями = with parents

Отчёты is the accusative plural form of отчёт and is the direct object of писать.

• отчёт = report
• отчёты = reports

Because отчёт is an inanimate masculine noun, its accusative plural looks the same as the nominative plural:

• nominative plural: отчёты
• accusative plural: отчёты

So писать отчёты means to write reports.

Because the auxiliary будет applies to both infinitives.

So:

• моя подруга будет работать с детьми и писать отчёты

means:

• my friend will work with children and write reports

Russian often avoids repeating the auxiliary when it is shared by two verbs.
You could repeat it for emphasis, but normally you do not:

• будет работать и писать = natural
• будет работать и будет писать = possible, but more repetitive

Because Russian usually does not need subject pronouns when the subject is already clear.

Here the subject is explicitly stated as:

• моя подруга

So there is no need to add она.

Russian often omits pronouns like я, ты, он, она, мы when the meaning is already clear from context or from the noun already given.