- There appear to be no functional systems in place to handle this.
This is correct.
We can see that "there" is not a noun in
"There is a dog in the garden" -> "There are two dogs in the garden."
There is not a noun phrase. It is an adverb. There appears in two forms:
Locative/demonstrative - Your pen is there = in/at that place
Existential: "There are pens in the cupboard. This is the weakened form and merely indicates the existence of the subject of the verb - "pens".
In "There appear to be no functional systems in place to handle this." The subject is "no functional systems", which is plural. The verb thus agrees with the subject = appear.
We can rewrite as "No functional systems appear to be in place to handle this."
The interesting point is if an adverb fronts a sentence the subject-verb order could be inverted, e.g. "Dearly did I love her."
In modern English this is now mainly restricted to locatives: "Here are your pens and there is the paper!"
OED:
I. As a demonstrative adverb.
- Expressing locality or position.
1.a. In or at that place; in the place (country, region, etc.) pointed to, indicated, or referred to, and away from the speaker; the opposite of here.
1850 J. McCosh Method Divine Govt. (ed. 2) ii. i. 142 Wherever we find law, there we see the certain traces of a lawgiver.
and
- Used unemphatically [...] the verb comes before its subject,[...] The same order was formerly observed after an introductory adverb or clause, [...] Grammatically, there is no difference between There comes the train! and There comes a time when, etc.; but, [...] in the latter it has been reduced to a mere anticipative element occupying the place of the subject which comes later.
Preceding or following a main verb, or following any verb, there, thus used, is stressless [...], but preceding be or an auxiliary, there has a slight stress, and the verb is enclitic (e.g. ˈthere-is, ˈthere-was, ˈthere-will).
1857 H. T. Buckle Hist. Civilisation Eng. I. vii. 399 From all these things there resulted consequences of vast importance.
4 d. especially with the verb to be: cf. be v. 1b, 4b there is, there are, are equivalent to French il est, il y a, German es ist, es sind, es gibt, Spanish hay. [...]
1608 W. Shakespeare King Lear vii. 459 For many miles about ther's not a bush.