Vom Wort her könnte es durchaus "in regelmäßigen Abständen" bedeuten, aber hier höre ich auch eher die Bedeutung "gelegentlich" - im Gegensatz zu frequently, häufig / oft.

periodically adverb - Definition, pictures, pronunciation and usage notes | Oxford Advanced Learner's Dictionary at OxfordLearnersDictionaries.com

periodically = regelmäßig immer wieder ... Das ist weniger oft als "frequently", aber im Gegensatz dazu systematisch und konsequent

I know that the dictionary definitions include the sense of "regularly"/"regelmäßig," but they also include the sense of "from time to time" -- and that is how "periodically" is being used here. Thus, the e-mails will not be read "systematisch und konsequent."

Re #5: The OP is about an (automatic) out-of-office reply and how often e-mails will be checked while the person is away, not how often they are out of the office.

Are you suggesting that "periodically" and "from time to time" are synonymous? Could that be an Americanism?

In fact, Merriam-Webster gives "from time to time" as one of two separate meanings, but, oddly, equates it with "frequently"

Periodically Definition & Meaning - Merriam-Webster

The bulk of dictionary definitions do not mention the sense of "once in a while" at all:

Periodic Definition & Meaning | Britannica Dictionary

PERIODICALLY | English meaning - Cambridge Dictionary

periodic | meaning of periodic in Longman Dictionary of Contemporary English | LDOCE (ldoceonline.com)

Are you suggesting that "periodically" and "from time to time" are synonymous?

I guess I'm saying that they can be used synonymously, not that they always are. (But what should I know? I've only been speaking English for over 60 years.).

Could that be an Americanism?

Could be. A native BE speaker would need to provide input as to such a usage takes place in BE.

AHD also includes the "from time to time" sense (and, unlike M-W, says nothing about "frequently"):

adj.

1. Having or marked by repeated cycles: periodic oscillations.

2. Happening or appearing regularly or from time to time: periodic dental checkups; periodic rain showers.

3. Characterized by periodic sentences.

Note that "regularly" and "from time to time" are separated by "or."

1.

 a. At regularly recurring or definite intervals. Now chiefly Science.

...

b. From time to time, occasionally.

In #0 steht:

In der Abwesenheitsnotiz eines Kunden können die Mitarbeiter angeben, wie Sie Ihren email-Account in der

Abwesenheit pflegen.

Nein, bitte. Korrektur für Eckibaer:  

In der Abwesenheitsnotiz eines Kunden können die Mitarbeiter angeben, wie sie ihren email-Account in der Abwesenheit pflegen.