tl;dr: It is likely that some Varangians, especially those who were on special duty in the imperial procession ceremony in Constantinople, were provided with the (blue?) silk cape that we can regard as 'uniform', but I cannot guarantee all the Varangians, especially stationed out of Constantinople, could access to the distribution of this kind of ceremonial crossing.

While Icelandic sagas sometimes mention luxurious weapons (like gilded with gold) and clothing of ex-Varangian Norse people, I find difficulty in finding any Greek source explicitly specifying their 'uniform'. I suppose many of these exotic 'Greek' equipment were granted by the emperor to them as a part of the shared booty, or tailored by their own order in the Byzantine empire.

I by came across the passage on the silk cape in Larsson's book, but so far no luck in identifying its original source (seemingly written in Greek primary text on the imperial parade). They probably also inherited their regiment ceremonial standards, that were to be held by their standard-bearer, Blöndal surmises (Blöndal and Benedkiz 1978: 23).

On the other hand, contemporary Greek (Byzantine) texts, such as Psellus (Psellos) and Anna Comnena, call the Varangians in common as 'those who carry their axe on their shoulder, [with shield]'. While they probably purchased and maintained these armaments at their own expense, these were the trademarks of the Varangians in Constantinople. The following were excerpts from them:

• 'Those then [the Varangian guards] were the warriors who rounded off that circles of shields, armed with long spears and single-edged battle-axes. The axes they carried on their shoulders, and with the spiked ends of the spears jutting oyt before and behind them the intervals between the ranks were, so to speak, roofed in' (Psellus on Michael VI, 1056-57, in: Sewter trans. 1966: 289).

• 'He [Alexios I] asked the men on guard at the various towers who they were. He learnt that at one point the defenders were the so-called 'Immotals' (a regiment peculiar to the Roman army); at another the Varangians from Thule (by these I mean the axe-bearing barbarians); at another the Nemitzi [the Germans] (there also belonged to a barbarian race which has for a long time served in the armed forces of the Empire).' (Alexiad II-9, in: Sewter trans. 1969: 95).

References:

• Anna Comnena. The Alkexiad, trans. E. R. A. Sewter. Harmondsworth: Penguin, 1969.
• Michael Psellus. Fourteen Byzantine Rulers, trans. E. R. A. Sewter. Harmondsworth: Penguin, 1966.

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• Blöndal, Sigfús. The Varangians of Byzantium, trans. & rev. Benedikt S. Benedikz. Cambridge: Cambridge UP, 1978 (pbk 2007).
• Larsson, Mats G. Väringer: Nordbor hos kejasaren i Miklagård. Stockholm, 1991.