Adjective describing someone who is in constant communication with someone else?

For this particular instance I like at [one's|your] disposal. You could also say that you are a [call|click] away.

I think you are really looking for accessible, approachable, available or responsive.

These, of course, are about being communicative when asked. If you are very communicative proactively, then I'd say you shouldn't be exactly advertising that, because it sounds like you'll be spamming me, or forcing your staying-in-touch with me upon me in other ways. I for one will then very much consider your "inferior" competitors who surely won't refuse answering a question or two of mine, either, but won't keep pestering me with their "just wanted to check back with you"s, "was wondering if everything's alright"s, and "our previous offer has expired, here's an update for your convenience"s.

Think about attentive and transparent.

The consistently-high level of communication you're describing is relative to a service rendered. Thus, make the words you seek to be contextual to that. it would have helped to call-out this more explicitly in the question's body.

Staying in constant contact in a service relationship is often regarded as being Attentive. Transparency reinforces this.

Perhaps linked (or linked up or linked with)

make, form, or suggest a connection with or between:

(with object):

rumors that linked his name with Judith

foreign and domestic policy are linked

(no object):

she was linked up with an artistic group

connect or join physically:

(with object):

a network of routes linking towns and villages

the cows are linked up to milking machines

(no object):

three different groups, each linking with the other